Abstract
We investigate the Born-Infeld--anti--de Sitter (BIAdS) black hole solutions in the four dimensions, which is a nonlinear generalization of the Reissner-Norstr\om-AdS (RNAdS) black hole. We analyze all thermodynamic quantities of the BIAdS black hole in the canonical ensembles, which are characterized by the charge $Q$, the mass $M$, the nonlinear parameter $b$, comparing with those of the RNAdS and Schwarzschild-AdS black hole. We find the forbidden region of $0\ensuremath{\le}bQl0.5$ for the presence of a charged BIAdS black hole. We also discuss the Hawking-Page phase transitions in the BIAdS black holes. Here we obtain a new Hawking-Page phase transition for the $bQ=0.5$ critical BIAdS black hole.
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