f(R, T, R_\mu\nu T^\mu\nu) gravity phenomenology and \Lambda CDM universe

We propose general f(R,T,R_{\mu\nu}T^{\mu\nu}) theory as generalization of covariant Horava-like gravity with dynamical Lorentz symmetry breaking. FRLW cosmological dynamics for several versions of such theory is considered. The reconstruction of the above action is explicitly done, including the numerical reconstruction for the occurrence of \Lambda CDM universe. De Sitter universe solutions in the presence of non-constant fluid are also presented. The problem of matter instability in f(R,T,R_{\mu\nu}T^{\mu\nu}) gravity is discussed.