Taking the nonextensive Tsallis and Rényi entropies into account, we explore thermodynamic properties and phase transitions of the Kerr Newman black holes (KNBH) in the microcanonical and canonical ensembles. We also compare our results with those obtained by attributing the Bekenstein-Hawking entropy bound to the mentioned black holes. Our analysis indicates that, similarly to the standard Boltzmann picture, isolated KNBH in the microcanonical approach are stable against axisymmetric perturbations in both Tsallis and Rnyi models. On the other hand, in considering the case when the black holes are enveloped by a bath of thermal radiation in the canonical treatment, the KNBH based on the Tsallis and Rényi entropies can be stable for some values of the entropy parameters, in contrast to the traditional Boltzmann framework. For the case of Rényi entropy, we nd that a Hawking-Page transition and a first order small black hole/large black hole transition can occur in a similar fashion as in rotating black holes in an anti-de Sitter space. Finally, we employ the Ruppeneir geometrothermodynamic technique to provide a new perspective on studying the nature of interactions between black hole microstructures, revealing a non-trivial
