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We investigate how the cosmological Equation of State can be used forscrutinizing extended theories of gravity, in particular, the Palatini $f(R)$gravity. Specifically, the approach consists, at first, in investigating theeffective Equation of State produced by a given model. Then, the inverseproblem can also be considered in view of determining which models arecompatible with a given effective Equation of State. We consider and solve somecases and show that, for example, power-law models are (the only models)capable of transforming barotropic Equations of State into effective barotropicones. 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 featurecapable of distinguishing Extended Gravity with respect to General Relativity.We also investigate quadratic and non-homogeneous effective Equations of Stateshowing, in particular, that they contain the Starobinsky model and other ones.

The effective Equation of State in Palatini $f(R)$ cosmology