law of thermodynamics. Similar considerations have been applied recently to describe the relative information entropy measure inside compact domains of an inhomogeneous universe [@CzM], where an explicit geometric model has been proposed to compute the $q$-parameter of the Rényi entropy in order to measure the effects of the gravitational entanglement problem.

In the case of black hole thermodynamics, we showed that large, asymptotically flat, Schwarzschild black holes can be in stable equilibrium with a thermal heath bath in the Rényi picture, and a Hawking–Page phase transition can occur in the system. This result offers a possible explanation for the problem of cosmic black hole nucleation in the early universe, and this mechanism might be the origin of large or super massive black holes that can be found in most galaxy centers. Many other interesting consequences can be deduced from the Rényi approach, but in this work we aim to achieve a more modest goal. In the present paper, by extending our previous investigations in the problem, we study the thermodynamic, stability and phase transition properties of Kerr black holes within the Rényi model and analyze whether similar results can be obtained to what we have found in the Schwarzschild case. 