In quantum gravity, the Hawking–Page phase transition is phase transition between AdS black holes with radiation and thermal AdS. Stephen Hawking and Don Page (Hawking & Page (1983)) showed that although AdS black holes can be in stable thermal equilibrium with radiation, they are not the preferred state below a certain critical temperature T C {\displaystyle T_{C}} . At this temperature, there will be a first order phase transition where below T C {\displaystyle T_{C}} , thermal AdS will become the dominant contribution to the partition function. The Hawking-Page phase transition between the unstable small black hole to stable large black hole phase is understood as a confinement-deconfinement phase transition in the dual conformal field theory via AdS/CFT correspondence.
References
Hawking, Stephen; Page, Don (1983), "Thermodynamics of black holes in anti-de Sitter space", Communications in Mathematical Physics, 87 (4): 577–588, Bibcode:1982CMaPh..87..577H, doi:10.1007/BF01208266, S2CID 53516597