  We find the general solution of the 6-dimensional Einstein-Gauss-Bonnet
equations in a large class of space and time-dependent warped geometries.
Several distinct families of solutions are found, some of which include black
string metrics, space and time-dependent solutions and black holes with exotic
horizons. Among these, some are shown to verify a Birkhoff type staticity
theorem, although here, the usual assumption of maximal symmetry on the horizon
is relaxed, allowing exotic horizon geometries. We provide explicit examples of
such static exotic black holes, including ones whose horizon geometry is that
of a Bergman space. We find that the situation is very different from
higher-dimensional general relativity, where Einstein spaces are admissible
black hole horizons and the associated black hole potential is not even
affected. In Einstein-Gauss-Bonnet theory, on the contrary, the non-trivial
Weyl tensor of such exotic horizons is exposed to the bulk dynamics through the
higher order Gauss-Bonnet term, severely constraining the allowed horizon
geometries and adding a novel charge-like parameter to the black hole
potential. The latter is related to the Euler characteristic of the
four-dimensional horizon and provides, in some cases, additional black hole
horizons.
