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We investigate the thermodynamics of hairy black holes in asymptoticallyAnti-de Sitter (AdS) space, including backreaction. Resorting to the Euclideanpath integral approach, we show that matter conformally coupled to Einsteingravity in five dimensions may exhibit a phase transition whose endpoint turnsout to be a hairy black hole in AdS5 space. The scalar field configurationhappens to be regular everywhere outside and on the horizon, and behavesasymptotically in such a way that respects the AdS boundary conditions that arerelevant for AdS/CFT. The theory presents other peculiar features in theultraviolet, like the existence of black holes with arbitrarily low temperaturein AdS5. This provides a simple setup in which the fully backreacting problemof a hair forming in AdS at certain critical temperature can be solvedanalytically.

The different phases of hairy black holes in AdS5 space