  We study inelastic neutrino scattering off hot nuclei for temperatures
relevant under supernova conditions. The method we use is based on the
quasiparticle random phase approximation extended to finite temperatures within
the thermo field dynamics (TQRPA). The method allows a transparent treatment of
upward and downward transitions in hot nuclei, avoiding the application of
Brink's hypothesis. For the sample nuclei $^{56}$Fe and $^{82}$Ge we perform a
detailed analysis of thermal effects on the strength distributions of allowed
Gamow-Teller (GT) transitions which dominate the scattering process at low
neutrino energies. For $^{56}$Fe and $^{82}$Ge the finite temperature
cross-sections are calculated by taking into account the contribution of
allowed and forbidden transitions. The observed enhancement of the
cross-section at low neutrino energies is explained by considering thermal
effects on the GT strength. For $^{56}$Fe we compare the calculated
cross-sections to those obtained earlier from a hybrid approach that combines
large-scale shell-model and RPA calculations.
