From thermodynamics to the solutions in gravity theory

In a recent work, we present a new point of view to the relation of gravity and thermodynamics, in which we derive the Schwarzschild solution through thermodynamic considerations by the aid of the Misner–Sharp mass in an adiabatic system. In this Letter we continue to investigate the relation between gravity and thermodynamics for obtaining solutions via thermodynamics. We generalize our studies on gravi-thermodynamics in Einstein gravity to modified gravity theories. By using the first law with the assumption that the Misner–Sharp mass is the mass for an adiabatic system, we reproduce the Boulware–Deser–Cai solution in Gauss–Bonnet gravity. Using this gravi-thermodynamic thought, we obtain a NEW class of solution in F(R)F(R) gravity in an n  -dimensional (n≥3n≥3) spacetime which permits three-type (n−2)(n−2)-dimensional maximally symmetric subspace, as an extension of our recent three-dimensional black hole solution, and four-dimensional Clifton–Barrow solution in F(R)F(R) gravity.