  In the presence of appropriate non-minimal couplings between a scalar field
and the curvature squared Gauss-Bonnet (GB) term, compact objects such as
neutron stars and black holes (BHs) can spontaneously scalarize, becoming a
preferred vacuum. Such strong gravity phase transitions have attracted
considerable attention recently. The non-minimal coupling functions that allow
this mechanism are, however, always postulated ad hoc. Here we point out that
families of such functions naturally emerge in the context of
Higgs--Chern-Simons gravity models, which are found as dimensionally descents
of higher dimensional, purely topological, Chern-Pontryagin non-Abelian
densities. As a proof of concept, we study spherically symmetric scalarized BH
solutions in a particular Einstein-GB-scalar field model, whose coupling is
obtained from this construction, pointing out novel features and caveats
thereof. The possibility of vectorization is also discussed, since this
construction also originates vector fields non-minimally coupled to the GB
invariant.
