We investigate how the cosmological Equation of State can be used for
scrutinizing extended theories of gravity, in particular, the Palatini f(R)
gravity. Specifically, the approach consists, at first, in investigating the
effective Equation of State produced by a given model. Then, the inverse
problem can also be considered in view of determining which models are
compatible with a given effective Equation of State. We consider and solve some
cases and show that, for example, power-law models are (the only models)
capable of transforming barotropic Equations of State into effective barotropic
ones. Moreover, the form of Equation of State is preserved (only) for f(R)=R,
as expected. In this perspective, modified Equations of State are a feature
capable of distinguishing Extended Gravity with respect to General Relativity.
We also investigate quadratic and non-homogeneous effective Equations of State
showing, in particular, that they contain the Starobinsky model and other ones.Comment: 19 page