  We show that the combination of cubic invariants defining five-dimensional
quasitopological gravity, when written in four dimensions, reduce to the
version of four-dimensional Einsteinian gravity recently proposed by Arciniega,
Edelstein & Jaime, that produces second order equations of motion in a FLRW
ansatz, with a purely geometrical inflationary period. We introduce a quartic
version of the four-dimensional Einsteinian theory with similar properties, and
study its consequences. In particular we found that there exists a region on
the space of parameters which allows for thermodynamically stable black holes,
as well as a well-defined cosmology with geometrically driven inflation. We
briefly discuss the cosmological inhomogeneities in this setup. We also provide
a combination of quintic invariants with those properties.
