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We consider FRW cosmology in $f(R)= R+ \gamma R^2+\delta R^3$ modifiedframework. The Palatini approach reduces its dynamics to the simplegeneralization of Friedmann equation. Thus we study the dynamics intwo-dimensional phase space with some details. After reformulation of the modelin the Einstein frame, it reduces to the FRW cosmological model with ahomogeneous scalar field and vanishing kinetic energy term. This potentialdetermines the running cosmological constant term as a function of the Ricciscalar. As a result we obtain the emergent dark energy parametrization from thecovariant theory. We study also singularities of the model and demonstrate thatin the Einstein frame some undesirable singularities disappear.

Emergence of dynamical dark energy from polynomial $f(R)$ theory in Palatini formalism