  Explosive astrophysical systems - such as supernovae or compact star binary
mergers - provide conditions where exotic degrees of freedom can be populated.
Within the covariant density functional theory of nuclear matter we build
several general purpose equations of state which, in addition to the baryonic
octet, account for $\Delta(1232)$ resonance states. The thermodynamic stability
of $\Delta$-admixed nuclear matter is investigated in the limiting case of
vanishing temperature for charge fractions $Y_Q=0.01$ and $Y_Q=0.5$ and wide
ranges of the coupling constants to the scalar and vector mesonic fields.
General purpose equation of state models with exotica presently available on
the \textsc{CompOSE} database are further reviewed; for a selection of them we
then investigate thermal properties for thermodynamic conditions relevant for
core-collapse supernovae and binary neutron star mergers. Modifications induced
by hyperons, $\Delta(1232)$, $K^-$, pions and quarks are discussed.
