No scalar-haired Cauchy horizon theorem in charged Gauss-Bonnet black holes

Recently, a "no inner (Cauchy) horizon theorem" for static black holes with non-trivial scalar hairs has been proved in Einstein-Maxwell-scalar theories and also in Einstein-Maxwell-Horndeski theories with the non-minimal coupling of a charged (complex) scalar field to Einstein tensor. In this paper, we study an extension of the theorem to the static black holes in Einstein-Maxwell-Gauss-Bonnet-scalar theories, or simply, charged Gauss-Bonnet (GB) black holes. We find that no inner horizon with charged scalar hairs is allowed for the planar (k=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=0$$\end{document}) black holes, as in the case without GB term. On the other hand, for the non-planar (k=+/- 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=\pm 1$$\end{document}) black holes, we find that the haired inner horizon can not be excluded due to GB effect generally, though we can not find a simple condition for its existence. As some explicit examples of the theorem, we study numerical GB black hole solutions with charged scalar hairs and Cauchy horizons in asymptotically anti-de Sitter space, and find good agreements with the theorem. Additionally, in an Appendix, we prove a "no-go theorem" for charged de Sitter black holes (with or without GB terms) with charged scalar hairs in arbitrary dimensions.