r_{+}^2(a^2 - r_{+}^2)}{r_{+}^2 +a^2},$$ and the heat capacity at constant angular momentum is $$C_J = T_H \left( \frac{\partial S_{BH}}{\partial T_H} \right)_J 
 = \frac{2 \pi (r_{+}^2 -a^2)(r_{+}^2 + a^2)^2}{3 a^4 + 6 r_{+}^2 a^2 - r_{+}^4}\ .$$ $C_\Omega$ and $C_J$ can be written in simpler forms if we normalize them by $r_{+}^2$, i.e. $$\frac{C_{\Omega}}{r_{+}^2} = - \frac{2 \pi (1 - h^2)}{h^2 + 1 }$$ and $$\frac{C_{J}}{r_{+}^2} =  \frac{2 \pi (1 - h^2 )(h^2 + 1)^2}{3 h^4 + 6 h^2 - 1},$$ were we also introduced the normalized rotation parameter $h$ [[@okamoto1990thermodynamical]]{} as $$h \equiv \frac{|a|}{r_{+}} .$$ The $r_{+}$ horizon radius exists only for $|a| \le M$, which corresponds to $0 \le h \le 1$. The $h=0$ value describes a Schwarzschild black hole, while the limiting value $h=1$ belongs to the extreme Kerr black hole case. The heat capacities as functions of $h$ are plotted on Fig. \[fig:heat\_capacity\_Kerr\]. It can be seen that $C_\Omega$ is negative for $ 0\le h<1$ and $C_J$ diverges at $h_c = \sqrt{\frac{2}{3}\sqrt{3} - 1}$, where a pole occurs. $C_J$ is negative for $h<h_c$ and positive for $h>h_c$ values. The heat capacities coincide at the limit values $h=0$ and 1.

![Plots of the heat capacities $C_J$ (red solid 