Abstract
Recently, a dual relation $T_0(n+1)=T_{HP}(n)$ between the minimum temperature ($T_0(n+1)$) black hole phase and the Hawking-Page transition ($T_{HP}(n)$) black hole phase in two successive dimensions was introduced in \cite{Wei:2020kra},
which was reminiscent of the AdS/CFT correspondence, as the Hawking-Page transition temperature could be treated as the temperature of the dual physical quantity on the boundary and the latter corresponds to the one in the bulk.
In this paper, we discuss the Hawking-Page transition and the dual relations in AdS black holes surrounded by dark energy in general dimensions.
Our findings reveal the occurrence of the Hawking-Page transition observed
between the thermal AdS radiation and thermodynamically stable large AdS black holes, 
both in the spacetime surrounded by phantom dark energy 
and the spacetime surrounded by quintessence dark energy.
We discuss the effects of the phantom dark energy and quintessence dark energy on the Hawking-Page transition temperature.
Especially for the dual relation,
it works well for the case of the AdS black holes surrounded by phantom dark energy.
While for the case of the AdS black holes surrounded by quintessence dark energy, 
the dual relation should be modified under an open assumption that the state parameter and the density parameter of the quintessence dark energy depend on the dimensions of the spacetime.
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