  Without specifying a matter field nor imposing energy conditions, we study
Killing horizons in $n(\ge 3)$-dimensional static solutions in general
relativity with an $(n-2)$-dimensional Einstein base manifold. Assuming linear
relations $p_{\rm r}\simeq\chi_{\rm r} \rho$ and $p_2\simeq\chi_{\rm t} \rho$
near a Killing horizon between the energy density $\rho$, radial pressure
$p_{\rm r}$, and tangential pressure $p_2$ of the matter field, we prove that
any non-vacuum solution satisfying $\chi_{\rm r}<-1/3$ ($\chi_{\rm r}\ne -1$)
or $\chi_{\rm r}>0$ does not admit a horizon as it becomes a curvature
singularity. While many exact solutions with horizons are known for $\chi_{\rm
r}=-1$, we construct asymptotic solutions near a Killing horizon for $\chi_{\rm
r}\in[-1/3,0)$ and show that a matter field is absent on the horizon for
$\chi_{\rm r}\in(-1/3,0)$. In contrast, there exists a matter on the horizon
for $\chi_{\rm r}=-1/3$, which is of the Hawking-Ellis type II and may be
interpreted as a null dust fluid. Differentiability of the metric on the
horizon depends on the value of $\chi_{\rm r}$ and such solutions can be
attached to the Schwarzschild-Tangherlini-type vacuum solution at the Killing
horizon in a $C^{1,1}$ regular manner without a lightlike thin-shell.
Additionally, some of those results are generalized in Lovelock gravity with a
maximally symmetric base manifold.
