  We present an inclusive study of the thermal and nuclear properties for the
hot nuclear matter using relativistic-mean field theory. We examine the effects
of temperature on the binding energy, pressure, thermal index, symmetry energy,
and its derivative for the symmetric nuclear matter using temperature-dependent
relativistic mean-field formalism for the well-known G2$^{*}$ and recently
developed IOPB-I parameter sets. The critical temperature for the liquid-gas
phase transition in an asymmetric nuclear matter system has also been
calculated and collated with the experimentally available data. We investigate
the approach of the thermal index as a function of nucleon density in the wake
of relativistic and non-relativistic formalism. The computation of neutrino
emissivity through the direct Urca process for the supernovae remnants has also
been performed, which manifests some exciting results about the thermal
stabilization and evolution of the newly born proto-neutron star. The central
temperature and the maximum mass of the proto-neutron star have also been
calculated for different entropy values.
