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Don N. Page investigated equilibrium black-hole thermodynamics with Stephen Hawking. Their 1983 work showed that sufficiently large black holes in Anti-de Sitter spacetime can have positive heat capacity and coexist in stable thermal equilibrium with radiation.^[ref]

The canonical ensemble distinguishes local stability from global preference: a locally stable black-hole solution need not have lower free energy than the competing thermal state.

The four-dimensional Schwarzschild--anti-de Sitter black hole is a static, spherically symmetric vacuum spacetime with negative cosmological constant $\Lambda =-3/L^2$. In units $G=c=\hslash =k_B=1$, its metric tensor isHere $L$ is the Anti-de Sitter curvature length and $M>0$ is the mass parameter. Setting $f(r_{+})=0$ gives $M=r_{+}(1+r_{+}^2/L^2)/2$. The surface gravity $\kappa =f^{\prime }(r_{+})/2$ then gives the Hawking temperatureUnlike its asymptotically flat limit $L\to \infty$, its temperature eventually increases with horizon radius. Differentiation gives a minimum at $r_{+}=L/\sqrt{3}$, separating negative- and positive-heat capacity branches.^[ref]

Stephen Hawking studied how gravitation and quantum physics change thermal equilibrium. In his 1983 collaboration with Don N. Page, a negative cosmological constant altered the temperature–size relation of a black hole, permitting a large positive-heat capacity branch in Anti-de Sitter spacetime.^[ref]

This provided a concrete phase transition problem in black-hole thermodynamics, comparing the free energies of black-hole and thermal-radiation configurations under the same boundary conditions.