  We study the consistency of Scalar Gauss-Bonnet Gravity, a generalization of
General Relativity where black holes can develop non-trivial hair by the action
of a coupling $F(\Phi){\cal G}$ between a function of a scalar field and the
Gauss-Bonnet invariant of the space-time. When properly normalized,
interactions induced by this term are weighted by a cut-off, and take the form
of an Effective Field Theory expansion. By invoking the existence of a Lorentz
invariant, causal, local, and unitary UV completion of the theory, we derive
positivity bounds for $n$-to-$n$ scattering amplitudes including exchange of
dynamical gravitons. These constrain the value of all even derivatives of the
function $F(\Phi)$, and are highly restrictive. They require some of the scales
of the theory to be of Planckian order, and rule out most of the models used in
the literature for black hole scalarization.
