  The possibility of evading Lovelock's theorem at $d=4$, via a singular
redefinition of the dimensionless coupling of the Gauss-Bonnet term, has been
extensively discussed in the cosmological context. The term is added as a
quadratic contribution of the curvature tensor to the Einstein-Hilbert action,
originating theories of "Einstein Gauss-Bonnet" (EGB) type. These studies are
interlaced with those of the conformal anomaly effective action. We review some
basic results concerning the structure of these actions, their conformal
constraints around flat space and their relation to EGB theories. The local and
nonlocal formulations of such effective actions are illustrated. This class of
theories find applications in the seemingly unrelated context of topological
materials, subjected to thermal and mechanical stress.
