  Taking the nonextensive Tsallis and R\'enyi 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 R\'enyi 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 R\'enyi entropies can be stable for some values of the entropy
parameters, in contrast to the traditional Boltzmann framework. For the case of
R\'enyi entropy, we find 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 impact of nonextensive entropies.
