  This paper is devoted to the study of various aspects of projectable F(R)
Horava-Lifshitz (HL) gravity. We show that some versions of F(R) HL gravity may
have stable de Sitter solution and instable flat space solution. In this case,
the problem of scalar graviton does not appear because flat space is not vacuum
state. Generalizing the U(1) HL theory proposed in arXiv:1007.2410, we
formulate U(1) extension of scalar theory and of F(R) Horava-Lifshitz gravity.
The Hamiltonian approach for such the theory is developed in full detail. It is
demonstrated that its Hamiltonian structure is the same as for the
non-relativistic covariant HL gravity. The spectrum analysis performed around
flat background indicates towards the consistency of the theory because it
contains graviton with only transverse polarization. Finally, we analyze the
spatially-flat FRW equations for U(1) invariant F(R) Horava-Lifshitz gravity.
