  Vacuum energy density and stresses are investigated for a scalar field with
general curvature coupling parameter in (D+1)-dimensional de Sitter spacetime
with an arbitrary number of toroidally compactified spatial dimensions. The
corresponding expectation values are presented in the form of the sum of the
vacuum expectation values in uncompactified dS spacetime and the part induced
by the non-trivial topology. In the early stages of the cosmological evolution
the topological parts dominate. In this limit the behavior of the Casimir
densities does not depend on the curvature coupling parameter and coincides
with that for a conformally coupled massless field. At late stages of the
cosmological expansion the expectation values are dominated by the part
corresponding to uncompactified dS spacetime. The vanishing of the topological
parts is monotonic or oscillatory in dependence of the mass and the curvature
coupling parameter of the field.
