Schwarzschild--anti-de Sitter black hole (Schwarzschild-AdS black hole)

In four dimensions the Schwarzschild--anti-de Sitter black hole hasWriting the horizon radius as $r_{+}$ gives $M=r_{+}(1+r_{+}^2/L^2)/2$ and

The spherical five-dimensional Schwarzschild--anti-de Sitter metric has $f(r)=1+r^2/L^2-\mu /r^2$. Its horizon satisfies $\mu =r_H^2(1+r_H^2/L^2)$ and has area $2{\pi }^2{r}_H^3$.

The Schwarzschild--anti-de Sitter heat capacity has the sign of $d{T}_H/d{r}_{+}$. It is positive precisely for $r_{+}>L/\sqrt{3}$, so these large black holes are locally stable in a fixed-temperature reservoir.