  Two approaches to the study of cosmological density perturbations in modified
theories of Palatini gravity have recently been discussed. These utilise,
respectively, a generalisation of Birkhoff's theorem and a direct linearization
of the gravitational field equations. In this paper these approaches are
compared and contrasted. The general form of the gravitational lagrangian for
which the two frameworks yield identical results in the long-wavelength limit
is derived. This class of models includes the case where the lagrangian is a
power-law of the Ricci curvature scalar. The evolution of density perturbations
in theories of the type $f(R)=R-c /R^ b$ is investigated numerically. It is
found that the results obtained by the two methods are in good agreement on
sufficiently large scales when the values of the parameters (b,c) are
consistent with current observational constraints. However, this agreement
becomes progressively poorer for models that differ significantly from the
standard concordance model and as smaller scales are considered.
