description [@Hawking:1982dh]. According to this similarity, we also analyzed the question of a possible phase transition in the canonical picture and found that a Hawking–Page black hole phase transition occurs in a very similar fashion as in AdS space in the Boltzamann statistics. We showed that the corresponding critical temperature depends only on the $q$-parameter of the Rényi formula, just like it depends only on the curvature parameter in AdS space. For the stability analysis we considered both the Poincaré and the Hessian methods. The latter one could also be applied since the Rényi entropy is additive for composition (see the next section), and therefore the standard stability analysis is also reliable in this case. Both approaches confirmed the same stability results.

These findings might have some relevant consequences in black hole physics. In particular, if an effective physical model could be constructed on how to compute the $q$-parameter value for the Rényi entropy of black holes (or other strongly gravitating systems) in order to parametrize the non-local effects of the gravitational field, the Rényi statistics can provide a well behaving and additive entropy description of the system which is also compatible with the requirements of equilibrium and the zeroth 