  The equation of state of hot neutron star matter of n+p+e+$\mu$ composition
in $\beta$-equilibrium is studied for both neutrino-free isothermal and
neutrino-trapped isentropic conditions, using the formalism where the thermal
evolution is built upon its zero-temperature predictions in a self-consistent
manner. The accuracy of the parabolic approximation, often used in the finite
temperature calculation of hot neutron star matter, is verified by comparing
with the results obtained from the exact evaluation in the neutrino-free
neutron star matter. The equation of state of neutrino-trapped isentropic
matter at low entropic condition, relevant to the core-collapsing supernovae,
is formulated. In the isentropic matter, the particle fractions and equation of
state have marginal variance as entropy per particle varies between 1 to 3 (in
the unit of k$_B$), but the temperature profile shows marked variation. The
isentropes are found to be much less sensitive to the nuclear matter
incompressibility, but have a large dependence on the slope parameter L. The
bulk properties of the neutron stars predicted by the isentropic equation of
state for different entropy are calculated. A model calculation for the early
stage evolution of the protoneutron star to neutron star configuration is also
given.
