  We conducted a study on the thermal properties of stellar matter with the
nuclear energy density functional BCPM. This functional is based on microscopic
Brueckner-Hartree-Fock calculations and has demonstrated success in describing
cold neutron stars. To enhance its applicability in astrophysics, in this study
we extend the BCPM equation of state to finite temperature for $\beta$-stable
neutrino-free matter, taking into consideration the hot inner crust. Such an
equation of state holds significant importance for hot compact objects,
particularly those resulting from a binary neutron star merger event. Our
exploration has shown that with increasing temperature there is a fast decrease
of the crust-core transition density, suggesting that for hot stars it is not
realistic to assume a fixed value of this density. The microscopic calculations
also reveal that the presence of nuclear clusters persists up to $T=7.21$ MeV,
identified as the limiting temperature of the crust. Above this threshold, the
manifestation of clusters is not anticipated. Below this temperature, clusters
within the inner crust are surrounded by uniform matter with varying densities,
allowing for the distinction between the upper and lower transition density
branches. Moreover, we computed mass--radius relations of neutron stars,
assuming an isothermal profile for $\beta$-stable neutron star matter at
various temperature values. Our findings highlight the significant influence of
the hot inner crust on the mass--radius relationship, leading to the formation
of larger and more inflated neutron stars. Consequently, under our
prescription, the final outcome is a unified equation of state at finite
temperature.
