  We review the equation of state (EoS) models covering a large range of
temperatures, baryon number densities and electron fractions presently
available on the \textsc{CompOSE} database. These models are intended to be
directly usable within numerical simulations of core-collapse supernovae,
binary neutron star mergers and proto-neutron star evolution. We discuss their
compliance with existing constraints from astrophysical observations and
nuclear data. For a selection of purely nucleonic models in reasonable
agreement with the above constraints, after discussing the properties of cold
matter, we review thermal properties for thermodynamic conditions relevant for
core-collapse supernovae and binary neutron star mergers. We find that the
latter are strongly influenced by the density dependence of the nucleon
effective mass. The selected bunch of models is used to investigate the EoS
dependence of hot star properties, where entropy per baryon and electron
fraction profiles are inspired from proto-neutron star evolution. The
$\Gamma$-law analytical thermal EoS used in many simulations is found not to
describe well these thermal properties of the EoS. However, it may offer a fair
description of the structure of hot stars whenever thermal effects on the
baryonic part are small, as shown here for proto-neutron stars starting from
several seconds after bounce.
