  We study black holes in a modified gravity scenario involving a scalar field
quadratically coupled to the Gauss-Bonnet invariant. The scalar is assumed to
be in a spontaneously broken phase at spatial infinity due to a bare Higgs-like
potential. For a proper choice of sign, the non-minimal coupling to gravity
leads to symmetry restoration near the black hole horizon, prompting the
development of the scalar wall in its vicinity. The wall thickness depends on
the bare mass of the scalar and can be much smaller than the Schwarzschild
radius. In a weakly coupled regime, the quadratic coupling to the Gauss-Bonnet
invariant effectively becomes linear, and no walls are formed. We find
approximate analytical solutions for the scalar field in the test field regime,
and obtain numerically static black hole solutions within this setup. We
discuss cosmological implications of the model and show that it is fully
consistent with the existence of an inflationary stage, unlike the spontaneous
scalarization scenario assuming the opposite sign of the non-minimal coupling
to gravity. Our model predicts the speed of gravitational waves to be extremely
close to unity, - in a comfortable agreement with the observation of the
GW170817 event and its electromagnetic counterpart.
