  Thermodynamics of rotating black holes described by the R\'enyi formula as
equilibrium and zeroth law compatible entropy function is investigated. We show
that similarly to the standard Boltzmann approach, isolated Kerr black holes
are stable with respect to axisymmetric perturbations in the R\'enyi model. On
the other hand, when the black holes are surrounded by a bath of thermal
radiation, slowly rotating black holes can also be in stable equilibrium with
the heat bath at a fixed temperature, in contrast to the Boltzmann description.
For the question of possible phase transitions in the system, we show that a
Hawking-Page transition and a first order small black hole/large black hole
transition occur, analogous to the picture of rotating black holes in AdS
space. These results confirm the similarity between the R\'enyi-asymptotically
flat and Boltzmann-AdS approaches to black hole thermodynamics in the rotating
case as well. We derive the relations between the thermodynamic parameters
based on this correspondence.
