  We obtain an hybrid expression for the heat-kernel, and from that the density
of the free energy, for a minimally coupled scalar field in a Schwarzschild
geometry at finite temperature. This gives us the zero-point energy density as
a function of the distance from the massive object generating the gravitational
field. The contribution to the zero-point energy due to the curvature is
extracted too, in this way arriving at a renormalised expression for the energy
density (the Casimir energy density). We use this to find an expression for
other physical quantities: internal energy, pressure and entropy. It turns out
that the disturbance of the surrounding vacuum generates entropy. For $\beta$
small the entropy is positive for $r>2M$. We also find that the internal energy
can be negative outside the horizon pointing to the existence of bound states.
The total energy inside the horizon turns out to be finite but complex, the
imaginary part being interpreted as responsible for particle creation.
