  The current accelerated universe could be produced by modified gravitational
dynamics as it can be seen in particular in its Palatini formulation. We
analyze here a specific non-linear gravity-scalar system in the first order
Palatini formalism which leads to a FRW cosmology different from the purely
metric one. It is shown that the emerging FRW cosmology may lead either to an
effective quintessence phase (cosmic speed-up) or to an effective phantom
phase. Moreover, the already known gravity assisted dark energy dominance
occurs also in the first order formalism. Finally, it is shown that a dynamical
theory able to resolve the cosmological constant problem exists also in this
formalism, in close parallel with the standard metric formulation.
