In a cylindrical cholesteric droplet with air interface under a temperature gradient, we found that the director field showed a steady rotation, whose direction depended on temperature. The flow-field measurements with the fluorescence photo-bleaching method revealed the appearance of a convection and a rotational flow under the temperature gradient. The rotational flow showed a differential rotation, which is considered to be driven by the convection of an inhomogeneous flow. Moreover, the direction of the convection depended on the temperature as similar to that of the director rotation; it is indicated that the director rotation is also induced by the convection, as well as the rotational flow.