Exotic black hole thermodynamics in third-order Lovelock gravity

Lovelock gravity represents a large class of classical metric gravitational theories that all possess a number of desirable properties, namely, having Lagrangians that are manifestly diffeomorphism-invariant and lead to second-order equations of motion ⁵ . Such theories differ from general relativity (which is the unique Lovelock theory in four dimensions ⁶ ) through the inclusion of higher-curvature terms in the Einstein–Hilbert action, which ordinarily contribute to the equations of motion only in higher dimensions (when d > 4). The Lovelock Lagrangian is

$$\begin{equation} \mathcal{L}=\frac{1}{16 \pi{G}} \sum_{k=0}^{K} \hat{\alpha}_{k} \mathcal{L}^{(k)}\ , \end{equation} \tag{ 2.1 }$$

where $\hat{\alpha}_{k}$ are the Lovelock coupling constants and $\mathcal{L}^{(k)}$ are the Euler densities, which are constructed through various contractions of the Riemann tensor R_abcd as

$$\begin{equation} \mathcal{L}^{(k)}=\frac{1}{2^{k}}\, \delta_{c_{1} d_{1} \ldots c_{k} d_{k}}^{a_{1} b_{1} \ldots a_{k} b_{k}} \,{R_{a_{1} b_{1}}}^{\!c_{1} d_{1}} \ldots {R_{a_{k} b_{k}}}^{\!c_{k} d_{k}}\ . \end{equation} \tag{ 2.2 }$$

With $\mathcal{L}^{(0)} = 1$, $\hat{\alpha}_0 = -2\Lambda$, $\mathcal{L}^{(1)} = R$ and $\hat{a}_1 = 1$, the K = 1 truncation of (2.1) yields the Einstein–Hilbert action. Notably, the kth Euler density is a topological invariant when $d\leqslant2k$, and therefore does not contribute to the equations of motion of the resulting theory. Nontrivial deviations from general relativity therefore occur only when the dimension of spacetime is large enough for a given finite truncation of the sum (2.1), namely when $d\gt2K$.

The full d-dimensional action of Lovelock theory with matter can thus be written as

$$\begin{equation} S=\int \textrm d^{d} x \sqrt{-g}\left(\frac{1}{16 \pi{G}_{N}} \sum_{k=0}^{K} \hat{\alpha}_{k} \mathcal{L}^{(k)}+\mathcal{L}_\text{matter}\right)\ , \end{equation} \tag{ 2.3 }$$

whose variation with respect to the metric yields the equations of motion

$$\begin{equation} \sum_{k}^{K} \hat{\alpha}_{k} \mathcal{G}_{a b}^{(k)}=8 \pi{G}_{N} T_{a b}\ , \end{equation} \tag{ 2.4 }$$

where $T_{a b}$ is the stress-energy tensor and $\mathcal{G}^{(k)}_{ab}$ are the Lovelock tensors, given by

$$\begin{equation} \mathcal{G}_{ab}^{(k)}=-\frac{1}{2^{(k+1)}}\, g_{z a}\, \delta_{b e_{1} f_{1} \ldots e_{k} f_{k}}^{z c_{1} d_{1} \ldots c_{k} d_{k}}\, {R_{c_{1} d_{1}}}^{e_{1} f_{1}} \ldots {R_{c_{k} d_{k}}}^{e_{k} f_{k}}\ .\end{equation} \tag{ 2.5 }$$

We will be interested in both uncharged and U(1) charged black holes, the latter involving an additional electromagnetic term in the action: $\mathcal{L}_{\text{matter}} = -4 \pi{G}_{N} F_{a b} F^{a b}$. The resulting field equations are then

$$\begin{equation} \sum_{k=0}^{K} \hat{\alpha}_{(k)} \mathcal{G}_{a b}^{(k)}=8 \pi{G}_{N}\left(F_{a c} F_{b}^{c}-\frac{1}{4} g_{a b} F_{c d} F^{c d}\right).\end{equation} \tag{ 2.6 }$$

We will use the following spherically symmetric ansatz ⁷ for the metric and gauge field:

$$\begin{equation} \textrm ds^2 = {\texttt{g}}_{ij} {\textrm dy^i \textrm dy^j} + {\gamma}_{\alpha\beta}\textrm dx^\alpha \textrm dx^\beta =-f(r)\textrm dt^2 + \frac{\textrm dr^{\,2}}{f(r)} + r^{\,2} \textrm d\Sigma_{d-2}^2 \ , \quad F = \frac{Q}{r^{\,d-2}}\textrm dt\wedge \textrm dr\ . \end{equation} \tag{ 2.7 }$$

Latin indices are reserved for coordinates in the t − r plane while Greek indices refer to coordinates on the base manifold $\textrm d\Sigma_{d-2}^2$. The importance of this metric is how one defines $\gamma_{\alpha \beta}$, which determines the geometry of the transverse sections (including the horizon). In this work we take the base manifold with metric $\gamma_{\alpha \beta}$ to be a $(d-2)$ compact space not necessarily of constant curvature. The constant curvature case (corresponding to 'ordinary' Lovelock black holes) has been studied extensively [17], while the non-constant curvature case has only been examined in the context of Gauss–Bonnet gravity [33]. Inserting the metric (2.7) into the field equations produces two simplified expressions:

$$\begin{equation} \begin{split} \mathcal{G}_{j}^{i} & \equiv \frac{-(d-2) \delta_{j}^{i}}{2 r^{\,d-2}} \frac{\textrm d}{\textrm d r} \sum_{n=0}^{K}\left\{b_{n}\left(r^{\,d-2 n-1} A_{n}\left(\frac{-f(r)}{r^{\,2}}\right)\right)\right\}=8 \pi{G}_{N} T_{j}^{\,i}\ ,\\ \mathcal{G}^{\alpha}_{\beta} & \equiv \frac{- \delta^{\alpha}_{\beta}}{2 r^{\,d-3}} \frac{\textrm d^2}{\textrm dr^{\,2}} \sum_{n=0}^{K} \left\{b_{n}\left( r^{\,d-2 n-1} A_{n}\left(\frac{-f(r)}{r^{\,2}}\right) \right) \right\} = 8 \pi{G}_{N} T^{\,\alpha}_{\beta}\ , \end{split} \end{equation} \tag{ 2.8 }$$

where

$$\begin{equation} A_{n}\left(\frac{-f(r)}{r^{\,2}}\right) \equiv \sum_{k=n}^{K} \alpha_{k} \binom{k}{n} \left( \frac{-f(r)}{r^{\,2}}\right)^{k-n} \ , \end{equation} \tag{ 2.9 }$$

and the b_n are arbitrary constants which depend on the geometry of the base manifold $\textrm d\Sigma_{d-2}^2$ (see section 2.2). These field equations determine the metric function f(r), which is obtained as a solution to the following polynomial of degree K in f(r):

$$\begin{equation} \sum_{n=0}^{K} \frac{b_{n}}{r^{\,2n}} \left( \sum_{k=n}^{K} \alpha_{k} \binom{k}{n}\left(\frac{-f(r)}{r^{\,2}}\right)^{k-n}\right )=\frac{16 \pi{G}_{N} M}{(d-2) \Sigma_{d-2} r^{\,d-1}}-\frac{8 \pi{G}_{N} Q^{2}}{(d-2)(d-3)} \frac{1}{r^{2 d-4}}. \end{equation} \tag{ 2.10 }$$

Here M represents the ADM mass, and Q is the charge defined by

$$\begin{equation} Q=\frac{1}{2 \Sigma_{d-2}}\int \star\, F\ .\end{equation} \tag{ 2.11 }$$

Finally, α_k are the re-scaled Lovelock coupling constants defined by >

$$\begin{equation} \alpha_{0}=\frac{\hat{\alpha}_{(0)}}{(d-1)(d-2)} = \quad \alpha_{1}=\hat{\alpha}_{(1)}, \quad \alpha_{k}=\hat{\alpha}_{(k)} \prod_{n=3}^{2 k}(d-n) \quad \text {for } \quad k \geqslant 2\ . \end{equation} \tag{ 2.12 }$$

Note that constant curvature base manifolds have $b_n = \kappa^n$ with $\kappa\in\{-1,0,+1\}$, corresponding to negative, flat and positive curvature respectively. This can be shown by direct comparison of (2.10) with the familiar form of the constant curvature f(r) polynomial given by

$$\begin{equation} \sum_{k=0}^{K} \alpha_{k}\left(\frac{\kappa-f}{r^{\,2}}\right)^{k}=\frac{16 \pi{G}_{N} M}{(d-2) \Sigma_{d-2}^{(\kappa)} r^{\,d-1}}-\frac{8 \pi{G}_{N} Q^{2}}{(d-2)(d-3)} \frac{1}{r^{2 d-4}}\ , \end{equation} \tag{ 2.13 }$$

where M, Q and the coupling constants are as defined above, and $\Sigma_{d-2}^{(\kappa)}$ is the volume of the constant curvature compact space.

In what follows we take K = 3 in the above expressions, corresponding to third-order Lovelock gravity, and set $\alpha_{1} = 1$ in order to recover general relativity in the $\{\alpha_2,\alpha_3\}\rightarrow 0$ limit. Useful for analysing the structure of the resulting solutions is the Kretschmann scalar, a curvature invariant which for our general metric (2.7) can be written as

$$\begin{align} \mathcal{K}&= R^{abcd }R_{abcd }\\ &=\left(\frac{\textrm d^{2} f(r)}{\textrm d r^{\,2}}\right)^{2} + \frac{2(d-2)}{r^{\,2}}\left( \frac{\textrm d \,f(r)}{\textrm dr} \right) ^2 + \frac{2(d-2)(d-3)\, f(r)^2}{r^{\,4}} - \frac{4 R[\gamma] \,f(r)}{r^{\,4}} +\frac{\mathcal{K}[\gamma]}{r^{\,4}}\ .\nonumber \end{align} \tag{ 2.14 }$$

It is a function of both the metric function f(r) and the Ricci and Kretschmann scalars of the base manifold ($R[\gamma]$ and $\mathcal{K}[\gamma]$, respectively), and provides a coordinate-free measure of curvature which can be used to locate singularities and other pathologies in the spacetime.

The primary motivation of this work is to examine the role that horizon geometry plays in determining the thermodynamic features of black hole spacetimes. Ordinarily, Birkhoff's theorem in general relativity strictly limits the types of allowed horizon geometries. For a general spherically symmetric metric of the form

$$\begin{equation} \textrm d s^{2}=-f\,(r, t) \textrm d t^{2}+\frac{\textrm d r^{\,2}}{f\,(r, t)}+r^{\,2} \textrm d \Sigma_{d-2}^{2}\ , \end{equation} \tag{ 2.15 }$$

the transverse part of the metric $\textrm d \Sigma_{d-2}^{2}$ must be an Einstein manifold (one for which $R_{\mu\nu} = kg_{\mu\nu}$ in local coordinates for some constant k) with spherical symmetry ⁸ . In d = 4 there is only one possibility, namely $\mathcal{S}^2$, implying that all vacuum solutions with SO(3) symmetry and positive curvature are diffeomorphic to the maximal extension of the Schwarzschild solution in an open subset. In higher dimensions other Einstein manifolds are allowed, such as the Bohm metrics [34].

In Lovelock gravity the base manifold is constrained through a number of relations among its Lovelock tensors, as discussed in [31]. The metric function f(r) is a solution to the polynomial (2.10) with the additional requirement that

$$\begin{equation} \hat{\mathcal{G}}_{\beta}^{(n) \alpha}=-\frac{(d-3) !\, b_{n}}{2(d-2 n-3) !} \delta_{\beta}^{\alpha}\ , \end{equation} \tag{ 2.16 }$$

where $\hat{\mathcal{G}}_{\beta}^{(n) \alpha}$ are simply the Lovelock tensors of the base manifold. The constants b_n appear in the polynomial (2.10) defining the metric function and are arbitrary, though they depend on the geometry of the base manifold through (2.16).

When $d = 2K+1$ the transverse space must be of constant curvature [23, 35], so the first dimension in which the Kth Lovelock term begins to contribute only admits spherical, flat, and hyperbolic transverse sections and has $b_{n} = \kappa^{n}$ with $\kappa\in\{-1,0,+1\}$. For third-order Lovelock gravity, this means that only black hole solutions in $d\geqslant 8$ can possess exotic horizon geometries. Furthermore, we emphasise that significant restrictions on the allowed geometries arise from the field equations. For example, the Bohm metrics

$$\begin{equation} \textrm d s^{2}=\textrm d \rho^{2}+a(\rho)^{2} \textrm d \Omega_{p}^{2}+b(\rho)^{2} \textrm d \Omega_{q}^{2} \end{equation} \tag{ 2.17 }$$

represent an infinite family of local metrics that can be extended to manifolds of topology $S^{p+q+1}$. Black holes with Bohm horizons have non-constant curvature, yet of the infinitely many metrics given by different choices of $a(\rho)$ and $b(\rho)$ in (2.17), only one choice turns out to be admissible as a horizon due to the Gauss–Bonnet correction appearing when $K\geqslant 2$ [27]. Higher orders in Lovelock theory likely further constrain the allowed choices of horizon geometry, though this remains a currently unexplored feature of exotic black holes in higher-dimensional Lovelock gravity. In this work we focus on choices of b_n that produce everywhere real and analytic metric functions, ensuring that curvature scalars are finite and that the spacetime has the appropriate asymptotics, rather than attempting to derive general constraints on the b_n 's. We will see that many interesting features appear even in the limited parameter space we study herein.

Setting K = 3 in (2.10) gives the following polynomial for the metric function f:

$$\begin{align} -\frac{\alpha_{3} \,f^{\,3}}{r^{\,6}}\!+\!\left(\!\!\frac{\alpha_{2}}{r^{\,4}}+\frac{3 b_{1} \alpha_{3}}{r^{\,6}}\!\!\right) f^{\,2}\!\!&+\!\left(\!\!-\frac{1}{r^{\,2}}-\frac{2 b_{1} \alpha_{2}}{r^{\,4}}- \frac{3 b_{2} \alpha_{3}}{r^{\,6}}\!\!\right) f +\alpha_{0}+\frac{b_{1}}{r^{\,2}}+&\!\!\frac{b_{2} \alpha_{2}}{r^{\,4}}+\frac{b_{3} \alpha_{3}}{r^{\,6}}=A(r),\nonumber \\ A(r) &\equiv \frac{16 \pi M }{(d-2) \Sigma_{d-2} r^{\,d-1}}-\frac{8 \pi Q^{2} }{(d-2)(d-3)r^{\,2d-4}}. \end{align} \tag{ 4.1 }$$

This is a cubic equation for f possessing three distinct solutions. The first is given by

$$\begin{align} f_E=\frac { \Big( r^4\left(3 \alpha_3+\alpha_2^2\right)- 9\left(b_1^2-b_2\right) {\alpha_3}^{2} \Big)+\Big( r^{\,2}\alpha_2+3 \alpha_3 b_1\Big) \left(\tfrac{1}{2}\big(Y+\sqrt {X}\big)\right)^{1/3}-\Big( \tfrac{1}{2}\big(Y+\sqrt {X}\big) \Big) ^{2/3}}{3 \alpha_3\Big({\tfrac{1}{4}\big(Y+\sqrt {X}}\big)\Big)^{1/3}} \nonumber \\ \end{align} \tag{ 4.2 }$$

where we have defined

$$\begin{align*} X\equiv&\ \Big(27 \alpha_3^3 \big(2 b_1^3-3 b_1 b_2+b_3\big)-\big(27\alpha_3^2\big(A(r)-\alpha_0\big)-2 \alpha_2^3 +9 \alpha_2 \alpha_3 \big) r^6\Big)^2\nonumber \\ &\ +4 \Big(9 \alpha_3^2 \big(b_2-b_1^2\big)-\big(\alpha_2^2 +3 \alpha_3 \big)r^4\Big)^3\ ,\nonumber \\ Y\equiv&\ (81 b_1 b_2-54 b_1^3-27 b_3) \alpha_3^3+\Big(27\alpha_3^2 \big(A(r)-\alpha_0\big)+9 \alpha_2 \alpha_3 -2 \alpha_2^3\Big) r^6\nonumber\ .\end{align*}$$

The subscript E indicates that this branch gives the appropriate Einstein limit when successively taking the limits $\alpha_3\rightarrow0$ and $\alpha_2\rightarrow 0$. The other two (conjugate) branches are given by the solutions

$$\begin{align} f=\frac {\left(1\mp i\sqrt {3} \right) \Big( r^4\!\left(3 \alpha_3+\alpha_2^2\right)- 9\left( b_1^2-b_2 \right) \alpha_3^{2} \Big)+\left( r^{\,2}\alpha_2+3\alpha_3 b_1\right)\! \sqrt[3]{4\big(Y+\sqrt {X}\big)}+ \left(1\pm i\sqrt {3} \right) \left( \tfrac{1}{2}\big(Y+\sqrt {X}\big) \right) ^{2/3} }{3 \alpha_3\sqrt[3]{4\big(Y+\sqrt {X}}\big)}. \end{align}$$

These two solutions are distinguished by a lack of convergent $\alpha_2\rightarrow0$ limit. We further differentiate between the two by considering their Gauss–Bonnet ($\alpha_3\rightarrow0$) limits. The solution f_GB, given by the upper sign, converges in this limit and yields the appropriate metric function for the Schwarzschild–AdS black hole (when $\{b_1,b_2,b_3\} = \{\kappa,\kappa^2,\kappa^3\}$) and is therefore referred to as the Gauss–Bonnet branch. The solution f_L , given by the lower sign, has a divergent $\alpha_3\rightarrow0$ limit and exists only in third-order Lovelock gravity. We refer to this as the Lovelock branch.

A number of considerations arise from the form of the solutions (4.2) and (4.3). As discussed in [17], a bound on the couplings $\{\alpha_2,\alpha_3\}$ can be obtained by requiring that all three solutions to (4.1) are asymptotically AdS. This is determined by positivity of the discriminant of the cubic equation for f at large-r, which is equivalent to the following constraint on the couplings:

$$\begin{equation} 27 {\alpha_{0}}^{2} {\alpha_{3}}^{2}+4 \alpha_{0} \alpha_{2}^{3}-18 \alpha_{0} \alpha_{2} \alpha_{3}-{\alpha_{2}}^{2}+4 \alpha_{3} \leqslant 0. \end{equation} \tag{ 4.3 }$$

When (4.3) is satisfied, all solutions are asymptotically AdS and one can smoothly take the Einstein limit by appropriate choice of solution. In figure 1, we plot regions in the parameter space for which this condition is satisfied.

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The inequality (4.3) puts an upper (and sometimes lower) bound on the allowed pressure for certain values of the couplings, given by

$$\begin{equation} P_{\pm}=\frac{(d-1)(d-2)}{432 \pi }\frac{9 \alpha_{2} \alpha_{3} - 2\alpha_{2}^{3} \pm 2 \left( \alpha_{2}^{2}-3\alpha_{3}\right)^\frac{3}{2} }{\alpha_{3}^2}\ .\end{equation} \tag{ 4.4 }$$

Now introducing the following dimensionless variables

$$\begin{align} \alpha=\frac{\alpha_{2}}{\sqrt{\alpha_{3}}}, \quad p=4 \sqrt{\alpha_{3}} P\ ,r_{+}=v \alpha_{3}^{\frac{1}{4}},\!\! \quad T=\frac{t \alpha_{3}^{-\frac{1}{4}}}{d-2},\! \quad m=\frac{16 \pi M}{(d-2) \Sigma_{d-2}^{(\kappa)} \alpha_{3}^{\frac{d-3}{4}}},\! \quad Q=\frac{q}{\sqrt{2}} \alpha_{3}^{\frac{d-3}{4}} \nonumber \\ \end{align} \tag{ 4.5 }$$

the pressure can be expressed as

$$\begin{equation} p_{\pm}= \frac{(d-1)(d-2)}{108 \pi} \left( 9\alpha - 2 \alpha^{3} \pm 2\left(\alpha^{2} - 3 \right)^{\frac{3}{2}}\right)\ .\end{equation} \tag{ 4.6 }$$

This matches our expectations for the constant curvature case, since these bounds are sensitive only to the asymptotic structure of the solutions that are independent of the horizon geometry. We see that both branches terminate at the same values of $\alpha = \sqrt{3}$, below which the pressure becomes imaginary and the spacetime is no longer asymptotically AdS. We therefore limit our analysis to couplings for which $\alpha \geqslant \sqrt{3}$. There is also a narrow region in the parameter space where a minimum pressure exists along with a maximum; in this case when $\sqrt{3}\lt\alpha\lt2$. When $\alpha\geqslant 2$ there is no lower bound on the pressure.

For ordinary thermodynamic systems, the equation of state provides a closure relation between thermodynamic degrees of freedom, and is usually phenomenological in nature. For black holes, the equation of state can be constructed directly from (3.1) and (3.5). For K = 3, we have that

$$\begin{align} P=&\ \frac{(d-2) T}{4 r_+}-\frac{(d-2) (d-3)b_{1}}{16 \pi{r}_+^{\,2}}+\frac{(d-2) \alpha_{2} b_{1} T}{2 r_+^{\,3}} -\frac{(d-2) (d-5)\alpha_{2} b_{2}}{16 \pi{r}_+^{\,4}}\nonumber \\ &+ \frac{3(d-2) \alpha_{3} b_{2} T}{4 r_+^{\,5}}-\frac{(d-2)(d-7)\alpha_{3} b_{3}}{16 \pi{r}_+^{\,6}}+\frac{Q^{2}}{2 r_+^{\,2 d -4}}\ , \end{align} \tag{ 4.7 }$$

and in terms of the dimensionless variables defined above we have

$$\begin{align} p=&\ \frac{t}{v}-\frac{(d-2)(d-3) b_{1}}{4 \pi{v}^{2}}+\frac{2 \alpha b_{1} t}{v^{3}}-\frac{(d-2)(d-5) \alpha b_{2}}{4 \pi{v}^{4}}+\frac{3 b_{2} t}{v^{5}} \nonumber \\ & - \frac{(d-2)(d-7)b_{3}}{4 \pi{v}^{6}} + \frac{q^{2}}{v^{2d-4}}\ .\end{align} \tag{ 4.8 }$$

From the equation of state, critical points are easily identified as locations where

$$\begin{equation} \frac{\partial p}{\partial v}=0\ , \quad \frac{\partial^2 p}{\partial v^2}=0\ .\end{equation} \tag{ 4.9 }$$

The first of these conditions yields the critical temperature $t_{\textrm c}$:

$$\begin{equation} t_{\textrm c}=\frac{(d-2)\left(-4 v_{\textrm c}^{10-2 d} q^{2} \pi+b_{1}(d-3) v_{\textrm c}^{4}+2 \alpha b_{2}(d-5) v_{\textrm c}^{2}+3b_{3} (d-7)\right)}{2 \pi\left(6 \alpha b_{1} v_{\textrm c}^{2}+v_{\textrm c}^{4}+15 b_{2}\right) v_{\textrm c}}. \end{equation} \tag{ 4.10 }$$

The second condition determines the critical volume v_c as a solution to the following polynomial:

$$\begin{align} 0=&\ 4 \pi{v}_{\textrm c}^{10-2 d}\left[(2 d-5) v_{\textrm c}^{4}+6\alpha b_{1}(2 d-7) v_{\textrm c}^{2}+15 b_{2}(2 d -9)\right](d-2) q^{2}\\ & + (d-2) \left[ 6\alpha ((d-3)b_{1}^{2}-b_{2}(d-5))v_{\textrm c}^{6} - b_{1}(d-3)v_{\textrm c}^{8} - 45 b_{3} b_{2}(d-7)\right] \nonumber\\ & + (d-2) \left[6 \alpha\left(5 b_{2}^{2}(d-5) -9(d-7) b_{1} b_{3} \right) v_{\textrm c}^{2} - 3\left( b_{2}\left( 4(d-5)\alpha^{2} - 15 d + 45 \right)b_1\right.\right.\nonumber\\ & + \left.\left.5 b_{3}(d -7) \right)v_{\textrm c}^{4}\right].\nonumber \end{align} \tag{ 4.11 }$$

Once the critical volume and temperature have been found, the equation of state (4.8) determines the critical pressure $p_{\textrm c}$. As (4.11) is generally a high-degree polynomial in $v_\textrm c$, multiple solutions may exist. We restrict our analysis to choices of parameters for which $\{v_{\textrm c},t_{\textrm c},p_{\textrm c}\}$ are all positive and real. Additional restrictions on the parameter space arise from entropic considerations, namely the positivity of (3.10) which implies that

$$\begin{equation} (d-6)(d-4) v^{4}+2 b_{1} \alpha(d-2)(d-6)v^{2}+ 3 (d-2)(d-4) b_{2} > 0\ .\end{equation} \tag{ 4.12 }$$

Since $d \geqslant 7$ and α > 0, this is automatically satisfied when $b_1\geqslant 0$ and $b_2\geqslant 0$.

The Gibbs free energy G and its dimensionless counterpart g can likewise be constructed. The equilibrium state of the system globally minimises the Gibbs free energy, and coexistence phases are easily identified as locations where $r_+(G)$ is multivalued. They are defined, respectively, as

$$\begin{equation} G=M-TS\ ,\qquad g=\frac{1}{\Sigma_{d-2}} \alpha_{3}^{\frac{3-d}{4}} G\ , \end{equation} \tag{ 4.13 }$$

giving

$$\begin{align} g =&\ \frac{q^{2}\!\left((d-4)\left(\left(v^{4}+3 b_{2}\right) d^{2}-\left(\frac{17 v^{4}}{2}+\frac{39 b_{2}}{2}\right)\! d+15 v^{4}+27 b_{2}\right)+2 b_{1}\!\left(d-\frac{7}{2}\right)(d-2) \alpha\, v^{2}(d-6)\right) }{2(d-3)(d-2)(d-4)\left(2 \alpha v^{2} b_{1}+v^{4}+3 b_{2}\right)(d-6)v^{d-3}} \nonumber\\ & -\frac{p\left(15 b_{2}(d-2)(d-4) v^{d-1}+(d-6)\left(6 b_{1} \alpha(d-2) v^{d+1}+v^{d+3}(d-4)\right)\right)}{4(d-6)(d-2)(d-4)\left(2 \alpha v^{2} b_{1}+v^{4}+3 b_{2}\right)(d-1)}\nonumber \\ & + \frac{3 b_{2} b_{3}(d-2)(d-4) v^{d-6}+6\left(b_{3}(d-6) b_{1}-\frac{b_{2}^{2}(d-4)}{2}\right) \alpha(d-2) v^{d-4}}{16\pi(d-6) v(d-4) \left(2 \alpha v^{2} b_{1}+v^{4}+3 b_{2}\right)} \nonumber\\ & + \frac{\left((d-2) b_{2}\left(36-9 d\right) b_{1}+5 b_{3}(d-4)(d-6)+2(d-2) b_{2}\left(\alpha^{2} d-6 \alpha^{2}\right) b_{1}\right) v^{d-2}}{16\pi(d-6) v(d-4) \left(2 \alpha v^{2} b_{1}+v^{4}+3 b_{2}\right)}\nonumber \\ & - \frac{\alpha\left(2(d-2) b_{1}^{2}-3\, b_{2}(d-4)\right) v^{d}-b_{1}(d-4) v^{d+2}}{16\pi v(d-4) \left(2 \alpha v^{2} b_{1}+v^{4}+3 b_{2}\right)}\ . \end{align} \tag{ 4.14 }$$

This can be plotted parametrically against the equilibrium temperature of the system, given by

$$\begin{align} t=\frac{-4 v^{10-2 d} q^{2} \pi+4 v^{6} p \pi+b_{1}(d-2)(d-3) v^{4}+\alpha b_{2}(d-2)(d-5) v^{2}+b_{3}(d-2)(d-7)}{4 \pi\left(2 \alpha v^{2} b_{1}+v^{4}+3 b_{2}\right) v} \nonumber \\ \end{align} \tag{ 4.15 }$$

and using the horizon radius $r_+$ as the parameter. The black hole size thus serves as the order parameter characterizing distinct equilibrium states and/or phases of the system. One can verify that these quantities agree with known results for spherical and hyperbolic black holes with constant curvature horizons [17]. Our analysis will be predicated on the free energy and equation of state, which provide complementary information about the location and nature of critical points of the black hole system. We will see how moving past the restriction of constant curvature transverse spaces markedly alters the nature of the transitions present in these systems.

We begin with the uncharged solutions in seven dimensions. With q = 0 the equation of state becomes

$$\begin{equation} p=\frac{t}{v}-\frac{5 b_{1}}{\pi{v}^{2}}+\frac{2 \alpha b_{1} t}{v^{3}}-\frac{5 \alpha }{2 \pi{v}^{4}}+\frac{3 t}{v^{5}}\ , \end{equation} \tag{ 5.1 }$$

and critical points as determined by (4.9) occur when

$$\begin{equation} t_{\textrm c}=\frac{10\left(b_{1} v_{\textrm c}^{2}+\alpha \right) v_{\textrm c}}{\pi\left(6 \alpha b_{1} v_{\textrm c}^{2}+v_{\textrm c}^{4}+15 \right)}\ ,\quad 10 b_{1} v_{\textrm c}^{6}+60 \alpha^{2} b_{1} v_{\textrm c}^{2}-30 \alpha v_{\textrm c}^{4}-450 b_{1} v_{\textrm c}^{2}-150 \alpha=0\ . \end{equation} \tag{ 5.2 }$$

We first examine the case when $b_{1} = b_{2} = b_3 = 1$, where the above reduces to

$$\begin{equation} t_{\textrm c}=\frac{10\left(v_{\textrm c}^{2}+\alpha\right) v_{\textrm c}}{\pi\left(v_{\textrm c}^{4}+6 \alpha v_{\textrm c}^{2}+15\right)}\ , \quad 10 v_{\textrm c}^{6}-30 \alpha v_{\textrm c}^{4}-\left(450 - 60 \alpha^{2}\right) v_{\textrm c}^{2}-150 \alpha=0\ . \end{equation} \tag{ 5.3 }$$

The critical volume in seven dimensions is therefore given by the solution to a cubic function in $v_{\textrm c}^2$. Only one solution is real and positive for $\alpha \geqslant \sqrt{3}$. Positivity of the entropy (3.10) will be automatically satisfied for black holes of any size since all coefficients are positive. In what follows, we fix the value of α to limit the size of the parameter space, noting where variations of α lead to important qualitative changes in the phase structure. Otherwise, changing α simply changes the numerical values of the various critical points and bounds we find herein. We set α = 3 so that we lie in the region of figure 1 where all solutions for the metric function f are real and positive, and a smooth Einstein limit exists. We are bounded only by a maximum pressure $p_+$, while the minimum pressure is null:

$$\begin{equation} p_+=\frac{5(4 \sqrt{6}-9)}{6 \pi} \approx 0.211\ , \quad p_-=0\ .\end{equation} \tag{ 5.4 }$$

In figure 2 we show the equation of state p(v), the Gibbs free energy g(t), and the phase diagram for these black holes. In figure 2(a), the equation of state is shown for temperatures above, below, and equal to the critical temperature $t_\textrm c$. Below $t_\textrm c$, we observe Van der Waals oscillations characteristic of a small-large black hole phase transition. States in the negative pressure region are not only thermodynamically unstable (since $\partial p/\partial v\gt0$), but also correspond to asymptotically de Sitter configurations (since $\Lambda\gt0$). The physical state of the system instead follows the coexistence line as determined by Maxwell's equal area law, and is always positive in pressure due to the long tail of the p(v) curve.

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In figure 2(b), we plot the Gibbs free energy as a function of equilibrium temperature at fixed pressure above, below, and at the critical pressure $p_\textrm c$, using $r_+$ as a parameter. The radiation phase (sometimes referred to as thermal AdS or empty anti-de sitter) corresponds to the g = 0 line. Along each curve, the black hole size increases as the curve is followed from left to right. As the temperature increases, the physical state of the system corresponds to whichever branch (including g = 0) globally minimises the free energy. Above the critical pressure (as in the red curve) we observe a Hawking–Page transition between the 'empty' radiation phase and a large black hole where the red line crosses g = 0. As the pressure decreases, a kink eventually forms at the critical pressure where a second-order phase transition between a small and large black hole occurs. Below $p_\textrm c$, a crossing forms where the small and large black hole branches are separated by an unstable intermediate branch, characterised by a negative specific heat. As the temperature increases, the system follows the lower curve and a first-order small-large transition occurs since the horizon size is discontinuous at the crossing.

A new feature appearing in our analysis is the presence of a triple point at sufficiently low pressure. As the pressure decreases towards zero the second-order phase transition occurring at the crossing within the swallowtail eventually intersects the radiation phase at g = 0, as seen in figure 3. At this novel triple point, the large and small black hole phases coexist with the radiation phase, with all three globally minimizing g. This is in contrast to an ordinary triple point which consists of three phases that all correspond to black hole states. The vertical line in figure 3 extends upwards and terminates at the maximum pressure $p_+$ (not shown). This novel triple point was also observed in Gauss–Bonnet black holes with non-constant curvature horizons [33], and has not previously been noted in studies of the phase structure of AdS black holes in third-order Lovelock gravity. Despite receiving significant attention in the hyperbolic ($\kappa = -1$) and flat (κ = 0) cases, notably in [35], the spherical κ = 1 case has not been analysed as thoroughly.

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Finally, we observe that the phase structure is highly sensitive to variations of α. As α increases, the critical pressure decreases, until α ≈ 3.85 where a turning point occurs and p_c begins to increase monotonically with α. When α ≈ 4.56 the critical pressure coincides with the maximum pressure $p_+$, while above this value, no critical points occur. Despite this, one can still examine phases with $\alpha\gtrsim4.56$, however in the uncharged case the phenomenology is limited to Hawking–Page phase transitions between radiation and a large black hole. When $\alpha \gtrsim 3.87$ the novel triple point no longer occurs at any allowed pressure. In this case the free energy at the small-large transition always lies above g = 0, and only a Hawking–Page transition occurs (for any value of p). Finally as the value of α increases, the triple point moves to the right, eventually merging with the critical point. Above this value of α only a Hawking–Page transition occurs. Henceforth we fix α = 3, as variations in α simply correspond to a rescaling of the b_n terms when $d\geqslant8$.

We now briefly examine charged solutions with positive-curvature transverse sections. When q ≠ 0 and $b_{1} = b_{2} = b_{3} = 1$, the critical temperature and volume are given by

$$\begin{equation} t_{\textrm c}={\frac {10{v_{{\textrm c}}}^{8}+10\alpha{v_{{\textrm c}}}^{6}-10\pi{q}^{2}}{\pi{v_{{\textrm c}}}^{5} \left( {v_{{\textrm c}}}^{4}+6\alpha{v_{{\textrm c}}}^{2}+15\right) }}\ , \end{equation} \tag{ 5.5 }$$

$$\begin{equation} v_{\textrm c}^{12}-3\alpha v_{\textrm c}^{10}+ \left( 6{\alpha}^{2}-45 \right) v_{\textrm c}^{8}-15\alpha{v_{{\textrm c}}}^{6}-(9\pi{v_{{\textrm c }}}^{4}+42\pi\alpha{v_{{\textrm c}}}^{2}+75\pi){q}^{2}=0\ . \end{equation} \tag{ 5.6 }$$

The critical pressure is given by

$$\begin{align} p_{\textrm c}={\frac {10{v_{{\textrm c}}}^{12}-5\alpha{v_{{\textrm c}}}^{10}+ \left( 10{ \alpha}^{2}-90 \right) {v_{{\textrm c}}}^{8}-15\alpha {v_{{\textrm c}}}^{6}-18\pi {q}^{2}{v_{{\textrm c}}}^{4}-28\pi\alpha{q}^{2}{v_{{\textrm c}}}^{2}-30\pi {q}^{2}}{2\pi{v_{{\textrm c}}}^{10} \left( {v_{{\textrm c}}}^{4}+6\alpha{v_{{\textrm c }}}^{2}+15 \right) }} \nonumber \\ \end{align} \tag{ 5.7 }$$

where $v_{\textrm c}$ will be the solution to the critical volume expression while the temperature has been replaced by the critical temperature. The critical volume equation can be re-expressed as the condition that $w_1 = w_2$ where we define

$$\begin{equation} w_{1}=\left(9 \pi{v}_{\textrm c}^{4}+42 \alpha \pi{v}_{\textrm c}^{2}+75 \pi\right) q^{2}, \quad w_{2}=v_{\textrm c}^{12}-3\alpha v_{\textrm c}^{10}+ \left( 6{\alpha}^{2}-45 \right) v_{\textrm c}^{8 }-15\alpha v_{\textrm c}^{6} . \end{equation} \tag{ 5.8 }$$

In this form the number of possible critical points can be easily determined through Descartes' rule of signs (graphically, we are concerned with the number of intersections between w₁ and w₂). w₁ is a quadratic in $v_{\textrm c}^2$ and no sign changes occur between terms, therefore no positive roots exist (q acts simply as a rescaling factor). For w₂ there are three sign changes, so a maximum of three positive roots exist and a minimum of one. For the case considered here where α = 3 there is only one positive root, therefore for any value of q there exists only one value of $v_{\textrm c}$ for which $w_{1} = w_{2}$, and only one critical point exists. In what follows the charge is fixed to q = 1 unless charge variations are being considered.

As with the uncharged solutions, we examine the thermodynamics behaviour through the equation of state and free energy, as shown in figure 4. On the right, the same oscillatory behaviour seen in the uncharged case is present, though the transition to the oscillatory region is now sensitive to the charge. As the value of charge is increased, the critical temperature decreases. In figure 4(b), the behaviour of the Gibbs free energy is qualitatively similar to the uncharged case, but with a different interpretation. In the canonical ensemble the physical charge is fixed, which places a lower bound on the allowed black hole size. In the case of d = 4 Einstein gravity for example, Reissner–Nördstrom black holes require that $m^2\gt q^2$ otherwise a naked singularity forms. This constraint generalises to higher-dimensional Lovelock black holes, placing a lower bound on the black hole size which depends on $\{\alpha,q,\Lambda\}$. Thus, Hawking–Page transitions do not occur since the radiation phase corresponds to an m = 0 black hole which cannot exist when q = 0. Instead, we observe a first-order phase transition between a small and large black hole when $p\lt p_\textrm c$, a second-order phase transition when $p = p_\textrm c$, and no transitions when $p\gt p_\textrm c$.

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Before moving on to eight dimensions, we make a brief comment about the hyperbolic case. When $b_{1} = -1,b_{2} = 1,b_{3} = -1$, there are no critical points for α = 3. Critical points are possible for values of α < 3 as seen in [17], however these values of α appear to correspond to spacetimes where there is no smooth limit between the three branches of solutions to (4.1), hence their properties and structures are not entirely understood. Furthermore, hyperbolic geometries contain discontinuities in both the Gibbs free energy and temperature making the thermodynamic analysis more difficult. We therefore limit our analysis to $b_1\gt0$ where such issues are not present.

In d = 8 with q = 0 and α = 3, the maximum and minimum pressures are

$$\begin{equation} p_{+}=\frac{7(-27+12 \sqrt{6})}{18 \pi} \approx 0.296\ ,\quad p_-=0\ , \end{equation} \tag{ 6.3 }$$

and the equation of state becomes

$$\begin{equation} p=\frac{t}{v}-\frac{15}{2 \pi{v}^{2}}+\frac{6 t}{v^{3}}-\frac{27 b_{2}}{2 \pi{v}^{4}}+\frac{3 b_{2} t}{v^{5}}-\frac{3 b_{3}}{2 \pi{v}^{6}}\ .\end{equation} \tag{ 6.4 }$$

The resulting critical volume and temperature are then given by

$$\begin{align} t_{\textrm c}=\frac{3\left(5 v_{\textrm c}^{4}+18 b_{2} v_{\textrm c}^{2}+3 b_{3}\right)}{\pi{v}_{\textrm c}\left(v_{\textrm c}^{4}+18 v_{\textrm c}^{2}+15 b_{2}\right)}\ ,\end{align} \tag{ 6.5 }$$

$$\begin{align} 15 v_{\textrm c}^{8}+\left(162 b_{2}-270\right) v_{\textrm c}^{6}+\left(297 b_{2}+45 b_{3}\right) v_{\textrm c}^{4}-\left(810 b_{2}^{2}-486 b_{3}\right) v_{\textrm c}^{2}+135 b_{2} b_{3}=0\ . \end{align} \tag{ 6.6 }$$

While the topological parameters are independent of each other, thermodynamic considerations lead to further constraints on their values. Re-arranging the equation of state (4.8) as

$$\begin{equation} t\rightarrow t(\,p,v,b_{2},b_{3})=\frac{2 \pi p v^{6}+15 v^{4}+27 v^{2} b_{2}+3 b_{3}}{2 \pi v\left(v^{4}+6 v^{2}+3 b_{2}\right)} \end{equation} \tag{ 6.7 }$$

reveals that negative values of b₂ lead to a divergent temperature due to the $(v^{4}+6 v^{2}+3 b_{2})$ term in the denominator. We therefore restrict our analysis to the case where $b_{2} \geqslant 0$. This $(v^{4}+6 v^{2}+3 b_{2})$ term is also present in the free energy g, which then diverges along with the temperature. From equation (6.6) we can determine the possible number of positive real $v_{\textrm c}$ solutions through Descartes' rule of signs, which are summarised in table 1 below.

Table 1. Possible number of positive real roots to 6.6 from Descartes' rule of signs applied to the topological parameters b₂ and b₃.

	$b_{2} = 0$	$b_{2}\gt0$
$b_{3} = 0$	1	3,1
$b_{3}\lt0$	1	3,1
$b_{3}\gt0$	2,0	4,2,0

We begin with the case where $b_{2} = 0$ and $b_{3} = -1$. This choice results in thermodynamic behaviour similar to the seven-dimensional case, with small-large transitions appearing generically below the critical temperature and pressure, as shown in figure 6. Note that there is no Hawking–Page transition at low temperatures, and that $G\rightarrow 0$ as $T\rightarrow 0$.

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Criticality is also present when $b_{2} = 0$ and $b_{3} = 0$. The free energy in this case is qualitatively similar to that of figure 6(b), and the p − v curves likewise follow figure 6(a). The critical pressure and temperature both increase as b₃ increases.

When $b_{2} = 0$ and $b_{3}\gt0$, two real and positive solutions for $v_{\textrm c}$ will exist for $b_{3}\lt\frac{324}{25}$. When $b_{3}\geqslant 1$, the novel triple point emerges (seen also in the uncharged seven-dimensional case above) with notable qualitative differences. Increasing b₃ likewise increases the temperature and pressure at which this triple point occurs. Eventually, when $b_{3} \approx 6$, the triple point is no longer present as the novel triple point merges with the smaller of the two critical points and only the Hawking–Page transition is seen.

Next we consider the case where $b_{2}\neq 0$. When $b_{2} = 1$ and $b_{3} = 0$, Van der Waals behaviour is observed corresponding to a small-large black hole transition, though no triple point occurs at any temperature/pressure. When $b_{3}\gt0$ instead, we observe the same qualitative behaviour noted above for $b_{2} = 0$ and $b_3\gt0$: the existence of a triple point below a certain value of the parameter (in this case $0\lt b_{3} \lt 0.8$) , and Hawking–Page transitions when $b_3\gt0.8$.

When d = 8 and q ≠ 0 the equation of state (4.8) becomes

$$\begin{equation} p=\frac{t}{v}-\frac{15}{2 \pi{v}^{2}}+\frac{6 t}{v^{3}}-\frac{27 b_{2}}{2 \pi{v}^{4}}+\frac{3 b_{2} t}{v^{5}}-\frac{3 b_{3}}{2 \pi{v}^{6}}+\frac{q^{2}}{v^{12}}\ , \end{equation} \tag{ 6.8 }$$

with the critical temperature and volume given by

$$\begin{equation} t_{\textrm c}=-\frac{3\left(-5 v_{\textrm c}^{10}-18 b_{2} v_{\textrm c}^{8}-3 b_{3} v_{\textrm c}^{6}+4 \pi q^{2}\right)}{\pi{v}_{\textrm c}^{7}\left(v_{\textrm c}^{4}+18 v_{\textrm c}^{2}+15 b_{2}\right)}\ , \end{equation} \tag{ 6.9 }$$

$$\begin{align} & 15 v_{\textrm c}^{14}+\left(162 b_{2}-270\right) v_{\textrm c}^{12}+\left(297 b_{2}+45 b_{3}\right) v_{\textrm c}^{10}-\left(810 b_{2}^{2}-486 b_{3}\right) v_{\textrm c}^{8}+135 b_{2} b_{3} v_{\textrm c}^{6}\nonumber\\ & \quad -\left(132 \pi{v}_{\textrm c}^{4}+1944 \pi{v}_{\textrm c}^{2}+1260 \pi b_{2}\right) q^{2}=0. \end{align} \tag{ 6.10 }$$

In principle, a systematic determination of the root structure of the polynomial above can be done using Descartes' rule of signs; however, the presence of b₃ and q-dependent terms means such an analysis quickly becomes cumbersome. Instead we proceed on a case-by-case basis, as only a few examples are required to demonstrate the salient features of charged d = 8 black holes.

As with the uncharged case, we begin with the spherical base manifold for which $b_{n} = 1$. In this case, we observe only a small-large transition terminating at a critical point $\{t_\textrm c,p_\textrm c\}$ for all values of the charge q. This behaviour is identical to what is shown in figure 4 for the d = 7 case. As q increases, $t_\textrm c$ and $p_\textrm c$ decrease, until they eventually achieve the q = 0 values shown in figure 6. That there can only be a small-large transition for 8-dimensional charged black holes is easily seen by separating the charge-dependent and charge-independent portions of (6.10), and studying the number of intersections of the resulting functions. Let

$$\begin{equation} F_1=F_2 \end{equation} \tag{ 6.11 }$$

with

$$\begin{equation} \begin{aligned} F_{1}&\equiv\left(132 \pi{v}_{\textrm c}^{4}+1944 \pi{v}_{\textrm c}^{2}+1260 \pi\right) q^{2}\\ F_{2}&\equiv15 v_{\textrm c }^{14}-108 v_{\textrm c}^{12}+342 v_{\textrm c}^{10}-324 v_{\textrm c}^{8}+135 v_{\textrm c}^{6}\ .\end{aligned}\end{equation} \tag{ 6.12 }$$

Plotting F₁ and F₂ for varying values of q reveals that for any finite value of the charge there is a unique value of $v_\textrm c$ which satisfies (6.11), as displayed in figure 7.

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As in the uncharged case, we plot the equation of state p(v) and free energy g(t), searching for Van der Waals oscillations in the p − v plane and the formation of a swallowtail in the g − t plane. Note that even though critical points may occur above g = 0, the radiation phase is inaccessible to the system since the charge is fixed in the canonical ensemble, so no Hawking–Page transitions will occur in any of the cases we examine below. Figure 8 displays the thermodynamic behaviouror the case where all $b_n$ are equal, while figure 9 has unequal $b_{n}$. We describe the behaviour in each case below.

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Beginning with $b_{2} = 0$, when $b_{3} \leqslant 0$ we observe a single critical point and the corresponding Van der Waals-like transition between a large and small black hole, for all values of q. Only quantitative changes in the critical temperature and volume occur when q is varied. When $b_{3}\gt0$, a triple point may occur for very specific values of q and b₃. For example, when $b_{3} = 0.01$, the standard small-intermediate-large black hole triple point occurs when $0\lt q\lesssim 0.0069$. When $q\gtrsim 0.0069$ the triple point vanishes and only a small-large transition occurs. On the other hand, when the charge is fixed the existence of the triple point is lower bounded by the parameter b₃. For example, when q = 0.05 we must have $b_{3}\gt0.4$ for a triple point to occur. This is shown in figure 9.

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Next we consider the case where $b_{2}\gt0$. For any choice of $b_{2}\gt0$ there is no choice of q and $b_{3}\lt0$ that yields a triple point. The only phenomena observed are small-large phase transitions. This is also the case when $b_{3} = 0$. When $b_{3}\gt0$, we have the possibility of a small-intermediate-large triple point. As above, the presence of this triple point is highly sensitive to the values of q and b₂. If $b_{2} = 0.1$ and $b_{3} = 0.1$, the triple point only occurs for $0\lt q\lt0.006$. If b₃ is increased while b₂ is kept fixed, the triple point will occur at a larger value of q. Correspondingly, the location of the triple point occurs at a larger temperature $t_\textrm c$ and pressure $p_\textrm c$. Eventually, if both b₃ and q are sufficiently large, the triple point moves above the maximum allowed pressure p_max, and only a small-large transition is physically allowed. Therefore as long as the maximum pressure bound is not violated, a triple point can occur for any choice of $b_2\gt0$ and $b_3\gt0$ provided the black hole has the correct charge.

The behaviour for fixed b₃ and varying b₂ and q is qualitatively different. As a concrete example, take $b_{3} = 0.1$ and increase b₂ and q from 0.1 and 0 respectively. A triple point occurs as long as $0.1\lt b_{2}\lt0.2$ and $0\lt q\lesssim0.003$. If both $b_2\geqslant0.2$ and $q\gtrsim 0.003$ only a small-large transition occurs. Generically, when the charge decreases the temperature and pressure at which the triple point occurs will increase. In all cases, analytic solutions to the locations of the triple/critical points are not available, so the accuracy in the bounds presented are constrained by the choice of numerical solver and precision used.

The equation of state, free energy, and phase diagram in the presence of a triple point are shown in figure 9. The triple point occurs at the intersection of the small, intermediate, and large black hole branches of the g − t diagram (indicated also by a red dot in the phase diagram). At this point, three distinct black hole states with different sizes all globally minimise the free energy at the same temperature. Figure 9(c) contains the phase diagram, which displays new features compared to the uncharged case. The rightmost branch distinguishes the large and intermediate black hole phases and terminates at a critical point, beyond which the system is in a 'superfluid' phase where the two states are no longer distinguishable. The nearly vertical branch separates the intermediate and small black hole phases, and extends upward until it terminates at the maximum allowed pressure p_max. This is conjectured to always occur before any critical point forms regardless of parameter choice, though a lack of analytic solutions for the location of the critical points prevents us from excluding the possibility of a second critical point occurring below p_max. Above the critical temperature/pressure which terminates the intermediate-large transition, the two phases are no longer distinguishable (there is no discontinuity in the horizon size). However, we still observe a phase transition between this 'merged' intermediate-large black hole branch and the small black hole branch.