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In this work we present a new framework of the gravity sector by consideringthe extension $F(R,w)$, in which $R$ is the Ricci scalar and $w$ is theequation of state. Three different choices of function $F(R,w)$ areinvestigated under the Palatini formalism. The models appear equivalent to$F(R)$ models of gravity with effective momentum-energy tensors. For lineardependence of Ricci scalar in which $F(R,w)=k(w)R$, the model appearsequivalent to Einstein-Hilbert action with effective momentum-energy tensor.Recovering the minimal coupling case of the last choice does not faceJordan-Einstein frame ambiguities and exhibits natural alignments with generalrelativity results in the matter\text{/} radiation dominated eras. We discusssome astrophysical implications of the model by considering scalar fields asdominant matter forms. We show that the Higgs inflation could be saved withinthe $F(R,w)$ model. We suggest some future investigations exemplified byconstant-roll inflation and universe evolution for $F(R)=f(R)k(w)$ where $f(R)$represents the Starobinsky gravitational form. Using the model and comparing itwith pure $F(R)$ gravity, we provide preliminary indications of $F(R,w)$'simpact. As a final note, we suggest using the Polytropic equation of state infuture works to investigate $F(R,w)$.

$F(R,w)$ Gravity: A new gravity framework