angular momentum and $\lambda$. It has a local maximum and a local minimum when $\lambda J_0$ is smaller than some critical value. In this case three black holes with different masses can coexist at a given temperature. The $C_{JR}$ heat capacity is negative in the phase between the two local extrema and positive otherwise (see also Fig. \[fig:hc\_J0\] in Sec. \[sec:Stability\]). Above the critical value of $\lambda J_0$, the local extrema disappear and the Rényi temperature becomes a monotonically increasing function of the energy. From these properties one can expect a stability change and a thermodynamic phase transition of Kerr black holes in the canonical ensemble. We will investigate this question in the next section.

![Plots of the Rényi temperature against the mass-energy parameter of Kerr black holes at fixed $J = J_0$ with $\lambda J_0 = 0.05$, 0.04, 0.03, 0.02 and 0.01 parameter values starting from the top respectively. The bottom, bold curve belongs to the standard Hawking temperature of the black hole in the Boltzmann model.[]{data-label="fig:MT_Kerr"}](figure4.eps)

Stability analysis {#stab}
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Kaburaki [*et al.*]{} [@Kaburaki:1993ah] have shown that isolated Kerr black holes are stable against axisymmetric perturbations in the standard picture of black hole thermodynamics. They also investigated the 