  We study the thermodynamic stability of charged black holes in gauged
supergravity theories in D=5, D=4 and D=7. We find explicitly the location of
the Hawking-Page phase transition between charged black holes and the pure
anti-de Sitter space-time, both in the grand-canonical ensemble, where electric
potentials are held fixed, and in the canonical ensemble, where total charges
are held fixed. We also find the explicit local thermodynamic stability
constraints for black holes with one non-zero charge. In the grand-canonical
ensemble, there is in general a region of phase space where neither the anti-de
Sitter space-time is dynamically preferred, nor are the charged black holes
thermodynamically stable. But in the canonical ensemble, anti-de Sitter
space-time is always dynamically preferred in the domain where black holes are
unstable.
  We demonstrate the equivalence of large R-charged black holes in D=5, D=4 and
D=7 with spinning near-extreme D3-, M2- and M5-branes, respectively. The mass,
the charges and the entropy of such black holes can be mapped into the energy
above extremality, the angular momenta and the entropy of the corresponding
branes. We also note a peculiar numerological sense in which the
grand-canonical stability constraints for large charge black holes in D=4 and
D=7 are dual, and in which the D=5 constraints are self-dual.
