  In order to classify modified gravity models according to their physical
properties, we analyze the cosmological linear perturbations for f(R,G)
theories (R being the Ricci scalar and G, the Gauss-Bonnet term) with a
minimally coupled perfect fluid. For the scalar type perturbations, we identify
in general six degrees of freedom. We find that two of these physical modes
obey the same dispersion relation as the one for a non-relativistic de Broglie
wave. This means that spacetime is either highly unstable or its fluctuations
undergo a scale-dependent super-luminal propagation. Two other modes correspond
to the degrees of freedom of the perfect fluid, and propagate with the sound
speed of such a fluid. The remaining two modes correspond to the entropy and
temperature perturbations of the perfect fluid, and completely decouple from
the other modes for a barotropic equation of state. We then provide a concise
condition on f(R,G) theories, that both f(R) and R+f(G) do fulfill, to avoid
the de Broglie type dispersion relation. For the vector type perturbation, we
find that the perturbations decay in time. For the tensor type perturbation,
the perturbations can be either super-luminal or sub-luminal, depending on the
model. No-ghost conditions are also obtained for each type of perturbation.
