  We investigate the thermodynamical properties of black holes in (3+1) and
(2+1) dimensional Einstein gravity with a negative cosmological constant. In
each case, the thermodynamic internal energy is computed for a finite spatial
region that contains the black hole. The temperature at the boundary of this
region is defined by differentiating the energy with respect to entropy, and is
equal to the product of the surface gravity (divided by~$2\pi$) and the Tolman
redshift factor for temperature in a stationary gravitational field. We also
compute the thermodynamic surface pressure and, in the case of the (2+1) black
hole, show that the chemical potential conjugate to angular momentum is equal
to the proper angular velocity of the black hole with respect to observers who
are at rest in the stationary time slices. In (3+1) dimensions, a calculation
of the heat capacity reveals the existence of a thermodynamically stable black
hole solution and a negative heat capacity instanton. This result holds in the
limit that the spatial boundary tends to infinity only if the comological
constant is negative; if the cosmological constant vanishes, the stable black
hole solution is lost. In (2+1) dimensions, a calculation of the heat capacity
reveals the existence of a thermodynamically stable black hole solution, but no
negative heat capacity instanton.
