  We quantize a scalar field at finite temperature T in the background of a
classical black hole, adopting 't Hooft's ``brick wall'' model with generic
mixed boundary conditions at the brick wall boundary. We first focus on the
exactly solvable case of two dimensional space-time. As expected, the energy
density is integrable in the limit of vanishing brick wall thickness only for
T=T_H - the Hawking temperature. Consistently with the most general stress
energy tensor allowed in this background, the energy density shows a surface
contribution localized on the horizon. We point out that the usual divergences
occurring in the entropy of the thermal atmosphere are due to the assumption
that the third law of thermodynamics holds for the quantum field in the black
hole background. Such divergences can be avoided if we abandon this assumption.
The entropy density also has a surface term localized on the horizon, which is
open to various interpretations. The extension of these results to higher
space-time dimensions is briefly discussed.
