  The effect of temperature and density dependence of the asymmetric nuclear
matter properties is studied within the extended relativistic mean field (ERMF)
model, which includes the contribution from the self and mixed interaction
terms by using different parametrizations obtained by varying the neutron skin
thickness $\Delta$r and $\omega$-meson self-coupling ($\zeta$). We observed
that the symmetry energy and its slope and incompressibility coefficients
decrease with increasing temperatures up to saturation densities. The ERMF
parametrizations were employed to obtain a new set of equations of state (EOS)
of the protoneutron star (PNS) with and without inclusion of hyperons. In our
calculations, in comparison with cold compact stars, we obtained that the
gravitational mass of the protoneutron star with and without hyperons increased
by $\sim 0.4M_{\odot}$ and its radius increased by $\sim 3$km. Whereas in case
of the rotating PNS, the mass shedding limit decreased with increasing
temperature, and this suggested that the keplerian frequency of the PNS, at T =
10 MeV should be smaller by $14-20%$ for the EOS with hyperon, as compared to
the keplerian frequency of a cold compact star.
