  We present the novel finite-temperature FSU2H$^*$ equation-of-state model
that covers a wide range of temperatures and lepton fractions for the
conditions in proto-neutron stars, neutron star mergers and supernovae. The
temperature effects on the thermodynamical observables and the composition of
the neutron star core are stronger when the hyperonic degrees of freedom are
considered. We pay a special attention to the temperature and density
dependence of the thermal index in the presence of hyperons and conclude that
the true thermal effects cannot be reproduced with the use of a constant
$\Gamma$ law.
