Recently, a no inner (Cauchy) horizon theorem for static black holes with non-trivial scalar hairs has been proved in Einstein-Maxwell-scalar theories. In this paper, we study an extension of the theorem to the static black holes in Einstein-Maxwell-Horndeski theories. We study the black hole interior geometry for some exact solutions and find that the spacetime has a (space-like) curvature singularity where the black hole mass gets an extremum and the Hawking temperature vanishes. We discuss further extensions of the theorem, including general Horndeski theories from disformal transformations.
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