8 Thermodynamics of the Maxwell - Gauss - Bonnet anti - de Sitter Black Hole with Higher Derivative Gauge Corrections

The local and global thermal phase structure for asymptotically anti-de Sitter black holes charged under an abelian gauge group, with both Gauss-Bonnet and quartic field strength corrections, is mapped out for all parameter space. We work in the grand canonical ensemble where the external electric potential is held fixed. The analysis is performed in an arbitrary number of dimensions, for all three possible horizon topologies spherical, flat or hyperbolic. For spherical horizons, new metastable configurations are exhibited both for the pure Gauss-Bonnet theory as well as the pure higher derivative gauge theory and combinations thereof. In the pure GaussBonnet theory with negative coefficient and five or more spatial dimensions, two locally thermally stable black hole solutions are found for a given temperature. Either one or both of them may be thermally favored over the anti-de Sitter vacuum corresponding to a single or a double decay channel for the metastable black hole. Similar metastable configurations are uncovered for the theory with pure quartic field strength corrections, as well combinations of the two types of corrections, in three or more spatial dimensions. Finally, a secondary Hawking-Page transition between the smaller thermally favored black hole and thermal anti-de Sitter space is observed when both corrections are turned on and their couplings are both positive.