f(R,T,RμνTμν) gravity phenomenology and ΛCDM universe

We propose general f(R,T,RμνTμν) theory as generalization of covariant Hořava-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 ΛCDM universe. De Sitter universe solutions in the presence of non-constant fluid are also presented. The problem of matter instability in f(R,T,RμνTμν) gravity is discussed.