  f(Lovelock) gravities are simple generalizations of the usual f(R) and
Lovelock theories in which the gravitational action depends on some arbitrary
function of the corresponding dimensionally-extended Euler densities. In this
paper we study several aspects of these theories in general dimensions. We
start by identifying the generalized boundary term which makes the
gravitational variational problem well-posed. Then, we show that these theories
are equivalent to certain scalar-tensor theories and how this relation is
characterized by the Hessian of f. We also study the linearized equations of
the theory on general maximally symmetric backgrounds. Remarkably, we find that
these theories do not propagate the usual ghost-like massive gravitons
characteristic of higher-derivative gravities on such backgrounds. In some
non-trivial cases, the additional scalar associated to the trace of the metric
perturbation is also absent, being the usual graviton the only dynamical field.
In those cases, the linearized equations are exactly the same as in Einstein
gravity up to an overall factor, making them appealing as holographic toy
models. We also find constraints on the couplings of a broad family of
five-dimensional f(Lovelock) theories using holographic entanglement entropy.
Finally, we construct new analytic asymptotically flat and AdS/dS black hole
solutions for some classes of f(Lovelock) gravities in various dimensions.
