This study is dedicated to a separable F(R,T)-gravity related to the anisotropic matter to extract the equation of state for F(R,T)-gravity. In this research, we offer insight into calculating the density and pressure in the phantom barrier, stiff fluid, and matter-dominated eras, respectively. As demonstrated, a spacetime in F(R,T)-gravity full of anisotropic matter is a generalized quasi-Einstein spacetime. In addition, we gain the equation of state of Codazzi type, Ricci semi-symmetric and Ricci-pseudo symmetric anisotropic fluid spacetime in F(R,T)-gravity. We prove an anisotropic spacetime in F(R,T)-gravity endowed with Codazzi-type Ricci tensor is a Yang Pure spacetime and Robertson–Walker spacetime. Furthermore, we try to give out the energy constraints of Penrose’s singularity theorem for black holes in an anisotropic fluid spacetime in F(R,T)-gravity. Lastly, we study hyperbolic Ricci solitons on anisotropic fluid spacetime in F(R,T)-gravity endowed with a torse-forming vector field, and for steady hyperbolic Ricci soliton, we deduced the equation of state of anisotropic fluid spacetime in F(R,T)-gravity.