  It is shown that only particular kinds of matter (in terms of the "radial"
pressure to density ratio $w$) can coexist with Killing horizons in black-hole
or cosmological space-times. Thus, for arbitrary (not necessarily spherically
symmetric) static black holes, admissible are vacuum matter ($w=-1$, i.e., the
cosmological constant or some its generalization) and matter with certain
values of $w$ between 0 and -1, in particular, a gas of disordered cosmic
strings ($w=-1/3$). If the cosmological evolution starts from a horizon (the
so-called Null Big Bang scenarios), this horizon can co-exist with vacuum
matter and certain kinds of phantom matter with $w\geq -3$. It is concluded
that normal matter in such scenarios is entirely created from vacuum.
