  We drastically simplify the problem of linearizing a general higher-order
theory of gravity. We reduce it to the evaluation of its Lagrangian on a
particular Riemann tensor depending on two parameters, and the computation of
two derivatives with respect to one of those parameters. We use our method to
construct a D-dimensional cubic theory of gravity which satisfies the following
properties: 1) it shares the spectrum of Einstein gravity, i.e., it only
propagates a transverse and massless graviton on a maximally symmetric
background; 2) the relative coefficients of the different curvature invariants
involved are the same in all dimensions; 3) it is neither trivial nor
topological in four dimensions. Up to cubic order in curvature, the only
previously known theories satisfying the first two requirements are the
Lovelock ones: Einstein gravity, Gauss-Bonnet and cubic-Lovelock. Of course,
the last two theories fail to satisfy requirement 3 as they are, respectively,
topological and trivial in four dimensions. We show that, up to cubic order,
there exists only one additional theory satisfying requirements 1 and 2.
Interestingly, this theory is, along with Einstein gravity, the only theory
which also satisfies 3.
