hole is given by the Kerr solution $$\begin{aligned}
 ds^2 & = & -dt^2 + \frac{2 M r}{\Sigma} \left( dt -a \sin^2 \theta d \phi \right)^2 
 + \frac{\Sigma}{\Delta} dr^2 + \Sigma d \theta^2  \nonumber \\
 & &+ (r^2 +a^2) \sin \theta d \phi^2\ ,\end{aligned}$$ where $$\Sigma = r^2 + a^2 \cos^2 \theta , ~~~ \Delta = r^2 + a^2 - 2Mr .$$ Here, $M$ is the mass-energy parameter of the black hole and $a$ is its rotation parameter. The thermodynamic quantities of a Kerr black hole can be expressed in terms of its horizon radius $r_{+} = M + \sqrt{M^2 - a^2}$, which is defined by taking $\Delta = 0$. The Hawking temperature of the black hole horizon is $$T_H = \frac{1}{2\pi}\left[ \frac{r_{+}}{r_{+}^2 + a^2} - \frac{1}{2 r_{+}} \right],$$ the Bekenstein-Hawking entropy is $$S_{BH} = \pi (r_{+}^2 + a^2),$$ the angular momentum of the black hole is $$J = \frac{a}{2 r_{+}}(r_{+}^2 + a^2),$$ the angular velocity of the horizon is $$\Omega = \frac{a}{r_{+}^2 + a^2},$$ and the mass-energy parameter can also be written as $$M = \frac{r_{+}^2 + a^2}{2 r_{+}}.$$ The heat capacity at constant angular velocity is given by $$C_{\Omega} = T_H \left( \frac{\partial S_{BH}}{\partial T_H} \right)_\Omega 
 = \frac{2 \pi 