  We examine higher-curvature gravities whose FLRW configurations are specified
by equations of motion which are of second order in derivatives, just like in
Einstein gravity. We name these theories Cosmological Gravities and initiate a
systematic exploration in dimensions $D \geq 3$. First, we derive an instance
of Cosmological Gravity to all curvature orders and dimensions $D \geq 3$.
Second, we study Cosmological Gravities admitting non-hairy generalizations of
the Schwarzschild solution characterized by a single function whose equation of
motion is, at most, of second order in derivatives. We present explicit
instances of such theories for all curvature orders and dimensions $D \geq 4$.
Finally, we investigate the equations of motion for cosmological perturbations
in the context of generic Cosmological Gravities. Remarkably, we find that the
linearized equations of motion for scalar cosmological perturbations in any
Cosmological Gravity in $D\geq 3$ contain no more than two time derivatives. We
explicitly corroborate this aspect by presenting the equations for the scalar
perturbations in some four-dimensional Cosmological Gravities up to fifth order
in the curvature.
