  In this paper we study the vacuum quantum fluctuations of the stationary
modes of an uncharged scalar field with mass $m$ around a Schwarzschild black
hole with mass $M$, at zero and non-zero temperatures. The procedure consists
of calculating the energy eigenvalues starting from the exact solutions found
for the dynamics of the scalar field, considering a frequency cutoff in which
the particle is not absorbed by the black hole. From this result, we obtain the
exterior contributions for the vacuum energy associated to the stationary
states of the scalar field, by considering the half-summing of the levels of
energy and taking into account the respective degeneracies, in order to better
capture the nontrivial topology of the black hole spacetime. Then we use the
Riemann's zeta function to regularize the vacuum energy thus found. Such a
regularized quantity is the Casimir energy, whose analytic computation we show
to yield a convergent series. The Casimir energy obtained does not take into
account any boundaries artificially imposed on the system, just the nontrivial
spacetime topology associated to the source and its singularity. We suggest
that this latter manifests itself through the vacuum tension calculated on the
event horizon. We also investigate the problem by considering the thermal
corrections via Helmholtz free energy calculation, computing the Casimir
internal energy, the corresponding tension on the event horizon, the Casimir
entropy, and the thermal capacity of the regularized quantum vacuum, analyzing
their behavior at low and high temperatures, pointing out the thermodynamic
instability of the system in the considered regime, i.e., $mM\ll 1$.
