  We investigate the vacuum expectation values of the field squared and the
energy-momentum tensor for a massless scalar field with general curvature
coupling parameter in spatially flat Friedmann-Robertson-Walker universes with
an arbitrary number of toroidally compactified dimensions. When the comoving
lengths of the compact dimensions are short compared to the Hubble length, the
topological parts coincide with those for a conformal coupling. This limit
corresponds to the adiabatic approximation. In the opposite limit of large
comoving lengths of the compact dimensions, in dependence of the curvature
coupling parameter, two regimes are realized with monotonic or oscillatory
behavior of the vacuum expectation values.
