F(R) gravity's rainbow and its Einstein counterpart

Motivated by UV completion of general relativity with a modification of a geometry at high energy scale, it is expected to have an energy dependent geometry. In this paper, we introduce charged black hole solutions with power Maxwell invariant source in the context of gravity's rainbow. In addition, we investigate two classes of $F(R)$ gravity's rainbow solutions. At first, we study energy dependent $F(R)$ gravity without energy momentum tensor, and then we obtain $F(R)$ gravity's rainbow in the presence of conformally invariant Maxwell source. We study geometrical properties of the mentioned solutions and compare their results. We also give some related comments regarding to thermodynamical behavior of the obtained solutions and discuss thermal stability of the solutions.