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 gravity 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.

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• Alternative gravity theories
• Classical black holes
• General relativity