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

Equations of state for hot neutron stars -- II. The role of exotic particle degrees of freedom