Modified F(R) Horava-Lifshitz gravity: a way to accelerating FRW cosmology

We propose a general approach for the construction of modified gravity which is invariant under foliation-preserving diffeomorphisms. Special attention is paid to the formulation of a modified F(R) Horava-Lifshitz gravity (FRHL), whose Hamiltonian structure is studied. It is demonstrated that the spatially flat FRW equations of FRHL are consistent with the constraint equations. The analysis of de Sitter solutions for several versions of FRHL indicates that the unification of the early-time inflation with the late-time acceleration is possible. It is shown that a special choice of parameters for FRHL leads to the same spatially flat FRW equations as in the case of the traditional F(R)gravity. Finally, an essentially most general modified Horava-Lifshitz gravity is proposed, motivated by its fully diffeomorphism-invariant counterpart, with the restriction that the action does not contain derivatives higher than the second order with respect to the time coordinate.