$$C_{\Omega R} = \frac{2 \pi r_{+}^2(a^2 - r_{+}^2)}{r_{+}^2 + a^2 
 + \pi \lambda (a^4 - r_{+}^4 + 4 a^2 r_{+}^2)},$$ while the heat capacity at constant angular momentum takes the form $$C_{JR} = \frac{2 \pi (r_{+}^2 - a^2)(a^2 + r_{+}^2)^2}{3 a^4 +6 a^2 r_{+}^2 - r_{+}^4 
 + \pi \lambda (a^2 + r_{+}^2)(a^4 +6 r_{+}^2 a^2 +r_{+}^4)}.$$ $C_{\Omega R}$ and $C_{JR}$ can also be written in the simpler, normalized forms as before, i.e. $$\frac{C_{\Omega R}}{r_{+}^2} = \frac{2 \pi (h^2 -1)}{h^2 + 1 + \pi k (h^4+ 4 h^2 -1)},$$ and $$\frac{C_{JR}}{r_{+}^2} = - \frac{2 \pi (h^2 - 1)(h^2 + 1)^2}{3 h^4 + 6 h^2 - 1 
 + \pi k (h^2 + 1)(h^4 + 6 h^2 + 1)},$$ where we also introduced the parameter $$k = \lambda r_{+}^2.$$ The heat capacities change their sign depending on the parameter values as plotted on Fig. \[fig:heat\_capacity\].

![Phase diagram of Kerr black holes with Rényi entropy. On the dashed curve $C_{JR}$ diverges while on the dotted curve $C_{\Omega R}$ diverges. In region I, $C_{\Omega R}<0$ and $C_{JR} <0$, in region II, $C_{\Omega R}<0$ and $C_{JR} >0$, while in region III, $C_{\Omega R}>0$ and $C_{JR} >0$.[]{data-label="fig:heat_capacity"}](figure2.eps)

For fixed angular momentum, both the Rényi and the Boltzmann 