  Thermal effects for inelastic neutrino-nucleus scattering off even-even
nuclei in the iron region are studied. Allowed and first-forbidden
contributions to the cross sections are calculated within the quasiparticle
random phase approximation, extended to finite temperatures within the
Thermo-Field-Dynamics formalism. The GT$_0$ strength distribution at finite
temperatures is calculated for the sample nucleus $^{54}$Fe. The
neutral-current neutrino-nucleus inelastic cross section is calculated for
relevant temperatures during the supernova core collapse. The thermal
population of the excited states significantly enhances the cross section at
low neutrino energies. In agreement with studies using a large scale
shell-model approach the enhancement is mainly due to neutrino up-scattering at
finite temperatures.
