  Fermionic condensate and the vacuum expectation values of the energy-momentum
tensor are investigated for twisted and untwisted massive spinor fields in
higher-dimensional de Sitter spacetime with toroidally compactified spatial
dimensions. The expectation values are presented in the form of the sum of
corresponding quantities in the uncompactified de Sitter spacetime and the
parts induced by non-trivial topology. The latter are finite and the
renormalization is needed for the first parts only. Closed formulae are derived
for the renormalized fermionic vacuum densities in uncompactified
odd-dimensional de Sitter spacetimes. It is shown that, unlike to the case of
4-dimensional spacetime, for large values of the mass, these densities are
suppressed exponentially. Asymptotic behavior of the topological parts in the
expectation values is investigated in the early and late stages of the
cosmological expansion. When the comoving lengths of compactified dimensions
are much smaller than the de Sitter curvature radius, to the leading order the
topological parts coincide with the corresponding quantities for a massless
fermionic field and are conformally related to the corresponding flat spacetime
results. In this limit the topological parts dominate the uncompactified de
Sitter part and the back-reaction effects should be taken into account. In the
opposite limit, for a massive field the asymptotic behavior of the topological
parts is damping oscillatory.
