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Recently, a ``no inner (Cauchy) horizon theorem" for static black holes withnon-trivial scalar hairs has been proved in Einstein-Maxwell-scalar theoriesand also in Einstein-Maxwell-Horndeski theories with the non-minimal couplingof a charged (complex) scalar field to Einstein tensor. In this paper, we studyan extension of the theorem to the static black holes inEinstein-Maxwell-Gauss-Bonnet-scalar theories, or simply, charged Gauss-Bonnet(GB) black holes. We find that no inner horizon with charged scalar hairs isallowed for the planar (k=0) black holes, as in the case without GB term. Onthe other hand, for the non-planar (k=+1,-1) black holes, we find that thehaired inner horizon can not be excluded due to GB effect generally, though wecan not find a simple condition for its existence. As some explicit examples ofthe theorem, we study numerical GB black hole solutions with charged scalarhairs and Cauchy horizons in asymptotically anti-de Sitter space, and find goodagreements with the theorem. As a byproduct, we find a ``no-go theorem" forcharged de Sitter GB black holes with charged scalar hairs in arbitrarydimensions.

No Scalar-Haired Cauchy Horizon Theorem in Charged Gauss-Bonnet Black Holes