  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 Gauss-Bonnet 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.
