entropies of a Kerr black hole are monotonically increasing functions of the mass parameter (Fig. \[fig:MS\_Kerr\]). An important difference however, is that while the [standard Boltzmann entropy is asymptotically convex]{} (being proportional to $M^2$ as approaching the static Schwarzschild solution in the large $M$ limit), [[the Rényi entropy is asymptotically concave, since it increases only logarithmically.]{}]{}

![Plots of the Rényi entropy as a function of the mass-energy parameter at fixed $J = J_0$ for the parameter values $\lambda J_0 = 0.05$, 0.04, 0.03, 0.02 and 0.01 starting from the bottom curve respectively. The top, bold curve belongs to the standard Boltzmann entropy of the Kerr black hole.[]{data-label="fig:MS_Kerr"}](figure3.eps)

On Fig. \[fig:MT\_Kerr\], we also plotted the temperature-energy relations for fixed angular momentum $J = J_0$. As it is well known, there is a maximum temperature in the case of the standard Boltzmann approach. In the smaller mass (low entropy) region of the $T_H(M)$ curve, the heat capacity $C_J$ is positive, while at larger masses (high entropy) region it is negative. The two regions correspond to the phases of $h>h_c$ and $h<h_c$ on Fig. \[fig:heat\_capacity\_Kerr\]. In the Rényi approach, the $T_R(M)$ curve has different behavior depending on the actual parameter values of the 