  The equation of state (EoS) and composition of dense and hot
$\Delta$-resonance admixed hypernuclear matter is studied under conditions that
are characteristic of neutron star binary merger remnants and supernovas. The
cold, neutrino free regime is also considered as a reference for the
astrophysical constraints on the EoS of dense matter. Our formalism uses the
covariant density functional (CDF) theory successfully adapted to include the
full $J^P=1/2^+$ baryon octet and non-strange members of $J^P=3/2^+$ decouplet
with density-dependent couplings that have been suitably adjusted to the
existing laboratory and astrophysical data. The effect of $\Delta$-resonances
at finite temperatures is to soften the EoS of hypernuclear matter at
intermediate densities and stiffen it at high densities. At low temperatures,
the heavy baryons $\Lambda$, $\Delta^-$,$\Xi^-$, $\Xi^0$ and $\Delta^0$ appear
in the given order if the $\Delta$-meson couplings are close to those for the
nucleon-meson couplings. As is the case for hyperons, the thresholds of
$\Delta$-resonances move to lower densities with the increase of temperature
indicating a significant fraction of $\Delta$'s in the low-density subnuclear
regime. We find that the $\Delta$-resonances comprise a significant fraction of
baryonic matter, of the order of $10\%$ at temperatures of the order of several
tens of MeV in the neutrino-trapped regime and, thus, may affect the supernova
and binary neutron star dynamics by providing, for example, a new source for
neutrino opacity or a new channel for bulk viscosity via the direct Urca
processes. The mass-radius relation of isentropic static, spherically symmetric
hot compact stars is discussed.
