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Thermodynamics in f(R) theories of gravity with coupling between matter and geometry

The character of thermodynamics for theories of gravity with coupling between matter and geometry in the FLRW universe has been studied in this paper. The result shows that on the apparent horizon, both the temporal and spatial component of the Friedmann equations can be written as the form of the first law of thermodynamics in the conservative context, but some additional terms are appeared due to the matter-geometry coupling and the non-equilibrium in the system. Moreover, the derived generalized second law of thermodynamics can not be always held in our considering case, which is different from the one given in general relativity. The condition to protect it is obtained, when we consider results from observational data and numerical solutions.