  We consider gauge theories based on abelian $p-$forms on real compact
hyperbolic spaces. Using the zeta-function regularization method and the trace
tensor kernel formula, we determine explicitly an expression for the vacuum
energy (Casimir energy) corresponding to skew-symmetric tensor fields. It is
shown that the topological component of the Casimir energy for co-exact forms
on even-dimensional spaces, associated with the trivial character, is always
negative. We infer on the possible cosmological consequences of this result.
