We develop a new parametrization for generic departures from the ΛCDM model based on the Markov-Mukhanov action with non-minimal matter-gravity coupling. In this approach the cosmological constant Λ and the gravitational coupling G become dynamical with their values determined by the explicit form of the matter-gravity coupling. We use the framework to build a simple phenomenological model where dynamical dark energy is naturally obtained as a consequence of the departure of the theory from classical General Relativity at low densities. We show how the model’‘s free parameters can be constrained from observational data and interpret the tentative observations of’‘phantom crossing’’ by the DESI collaboration in view of the proposed formalism.