  For arbitrary static space-times, it is shown that an equilibrium between a
Killing horizon and matter is only possible for some discrete values of the
parameter $w = p_1/\rho$, where $\rho$ is the density and $p_1$ is pressure in
the direction normal to the horizon. In the generic situation of a simple
(non-extremal) horizon and the slowest possible density decrease near the
horizon, this corresponds to $w = -1/3$, the value known for a gas of
disordered cosmic strings. An admixture of "vacuum matter", characterized by
$w=-1$ and nonzero density at the horizon, is also admitted. This extends the
results obtained previously for static, spherically symmetric space-times. A
new feature as compared to spherical symmetry is that higher-order horizons can
exist in the absence of vacuum matter if the horizon is a surface of zero
curvature, which can occur, e.g., in cylindrically symmetric space-times.
