  Conditions for thermodynamic stability of asymptotically anti-de Sitter
rotating black holes in D-dimensions are determined. Local thermodynamic
stability requires not only positivity conditions on the specific heat and the
moment of inertia tensor but it is also necessary that the adiabatic
compressibility be positive. It is shown that, in the absence of a cosmological
constant, neither rotation nor charge is sufficient to ensure full local
thermodynamic stability of a black hole.
  Thermodynamic stability properties of anti-de Sitter Myers-Perry black holes
are investigated for both singly spinning and multi-spinning black holes.
Simple expressions are obtained for the specific heat and moment of inertia
tensor in any dimension. An analytic expression is obtained for the boundary of
the region of parameter space in which such space-times are thermodynamically
stable.
