line) and $C_\Omega$ (blue dotted line) against $h$. $C_J$ diverges at $h_c$ where a pole occurs.[]{data-label="fig:heat_capacity_Kerr"}](figure1.eps)

The Rényi entropy function of black holes can be obtained by taking the formal logarithm of the Bekenstein-Hawking entropy, which in our non-Boltzmannian approach follows the nonadditive Tsallis statistics. The physical meaning of the $\lambda$ parameter is connected to the nonextensive and nonlocal nature of the problem, and the Rényi entropy of a general black hole in this picture is given by $$S_R = \frac{1}{\lambda} \ln (1 + \lambda S_{BH}).$$ The zeroth law compatible Rényi temperature is then defined as $$T_R = \frac{1}{\partial S_R / \partial M} = T_H ( 1 + \lambda S_{BH} ).$$

For the case of a Kerr black hole, the Rényi entropy and the corresponding temperature take the forms $$\label{eq:Renyi_entropy}
 S_R = \frac{1}{\lambda} \ln ( 1 + \pi \lambda (r_{+}^2 + a^2) )$$ and $$\label{eq:Renyi_temperature}
 T_R = \frac{(1 + \pi \lambda (r_{+}^2 + a^2))(r_{+}^2 - a^2)}{4 \pi r_{+} (r_{+}^2 + a^2)}.$$ The heat capacities can be obtained as $$C_{R} = T_R \left( \frac{\partial S_R}{\partial T_R} \right) 
 = \frac{C_{BH}}{1 + \lambda(S_{BH} + C_{BH})},$$ where $C_{BH} = T_H \left( \frac{\partial S_{BH}}{\partial T_H} \right)$. The heat capacity at constant angular velocity is then 