  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.
