  The thermodynamic properties of Schwarzschild-anti-de Sitter black holes
confined within finite isothermal cavities are examined. In contrast to the
Schwarzschild case, the infinite cavity limit may be taken which, if suitably
stated, remains double valued. This allows the correspondence between
non-existence of negative modes for classical solutions and local thermodynamic
stability of the equilibrium configuration of such solutions to be shown in a
well defined manner. This is not possible in the asymptotically flat case.
Furthermore, the non-existence of negative modes for the larger black hole
solution in Schwarzschild-anti-de Sitter provides strong evidence in favour of
the recent positive energy conjecture by Horowitz and Myers.
