  We investigate to what extent the temperature dependence of the nuclear
symmetry energy can affect the neutronization of the stellar core prior to
neutrino trapping during gravitational collapse. To this end, we implement a
one-zone simulation to follow the collapse until beta equilibrium is reached
and the lepton fraction remains constant. Since the strength of electron
capture on the neutron-rich nuclei associated to the supernova scenario is
still an open issue, we keep it as a free parameter. We find that the
temperature dependence of the symmetry energy consistently yields a small
reduction of deleptonization, which corresponds to a systematic effect on the
shock wave energetics: the gain in dissociation energy of the shock has a small
yet non-negligible value of about 0.4 foe (1 foe = 10^51 erg) and this result
is almost independent from the strength of nuclear electron capture. The
presence of such a systematic effect and its robustness under changes of the
parameters of the one-zone model are significative enough to justify further
investigations with detailed numerical simulations of supernova explosions.
