In this analysis the turning point method is applied to the stability problem in both the microcanonical and canonical ensembles, and we will show that stability changes appear in the latter case, which suggests that a Hawking–Page transition and a first order small black hole/large black hole phase transition occur in the system, similar to the one observed for charged and rotating black holes in AdS space [@Chamblin:1999tk; @Chamblin:1999hg; @Caldarelli:1999xj; @Tsai:2011gv; @Altamirano:2014tva]. This result provides a correspondence between the Kerr–Rényi and the Kerr-AdS–Boltzmann pictures, analogous to the one we reported in the Schwarzschild problem [@Biro:2013cra; @Czinner:2015eyk].

The plan of the paper is as follows. In Sec. \[sec:Renyi\] we discuss the foundations and motivation of the Rényi approach arising from nonextensive thermodynamics to the black hole problem. In Sec. \[sec:Kerr\] we introduce the Kerr solution and calculate its thermodynamic quantities within the Rényi statistics. In Sec. \[sec:Stability\] we investigate the thermodynamic stability problem of Kerr black holes in the Rényi model by the Poincaré turning point method both in the microcanonical and canonical treatments. We also discuss the question of possible phase transitions in this section. In Sec. \[sec:Kerr-AdS\] the thermodynamic stability problem of Kerr-AdS black holes in the standard Boltzmann 