  Generalized quasi-topological gravities (GQTGs) are higher-curvature
extensions of Einstein gravity characterized by the existence of non-hairy
generalizations of the Schwarzschild black hole which satisfy
$g_{tt}g_{rr}=-1$, as well as for having second-order linearized equations
around maximally symmetric backgrounds. In this paper we provide strong
evidence that any gravitational effective action involving higher-curvature
corrections is equivalent, via metric redefinitions, to some GQTG. In the case
of theories involving invariants constructed from contractions of the Riemann
tensor and the metric, we show this claim to be true as long as (at least) one
non-trivial GQTG invariant exists at each order in curvature ---and extremely
conclusive evidence suggests this is the case in general dimensions. When
covariant derivatives of the Riemann tensor are included, the evidence provided
is not as definitive, but we still prove the claim explicitly for all theories
including up to eight derivatives of the metric as well as for terms involving
arbitrary contractions of two covariant derivatives of the Riemann tensor and
any number of Riemann tensors. Our results suggest that the physics of generic
higher-curvature gravity black holes is captured by their GQTG counterparts,
dramatically easier to characterize and universal. As an example, we map the
gravity sector of the Type-IIB string theory effective action in AdS$_5$ at
order $\mathcal{O}({\alpha^{\prime}}^3)$ to a GQTG and show that the
thermodynamic properties of black holes in both frames match.
