  Static black holes in general relativity modified by a linear scalar coupling
to the Gauss-Bonnet invariant always carry hair. We show that the same
mechanism that creates the hair makes it incompatible with a cosmological
horizon. Other scalar-tensor models do not have this problem when
time-dependence of the scalar provides a natural matching to cosmology.
Scalar-Gauss-Bonnet is particularly rigid and such a scenario does not help. An
extra operator makes the theory behave like the other models and the
cosmological horizon can be accommodated. The hair, however, is drastically
altered.
