  We study the thermal effects on the nuclear matter (NM) properties such as
binding energy, incompressibility, free symmetry energy and its coefficients
using NL3, G3 and IU-FSU parameter sets of relativistic mean-field models.
These models being consistent with the properties of cold NM, have also been
used to study the effect of temperature by incorporating the Fermi function.
The critical temperature for the liquid-gas phase transition in the symmetric
NM is found to be 14.60, 15.37 and 14.50 MeV for NL3, G3 and IU-FSU parameter
sets respectively, which is in excellent agreement with previous theoretical
and experimental studies. We inspect that the properties related to second
differential coefficient of the binding energy and free symmetry energy at
saturation density ( i.e. K 0 (n, T ) and Q sym,0) exhibit the contrary effects
for NL3 and G3 parameters as the temperature increases. We find that the
prediction of saturated curvature parameter ( K sym,0 ) for G3 equation of
state at finite temperature favour the combined analysis of K sym,0 for the
existence of massive pulsars, gravitational waves from GW170817 and NICER
observations of PSR J0030+0451. Further, we investigate the cooling mechanism
of newly born stars through neutrino emissivity controlled by direct Urca
process and instate some interesting remarks about neutrino emissivity. We also
deliberate the effect of temperature on the M-R profile of Proto-Neutron star.
