Thermodynamics of f(R) Gravity with Disformal Transformation

We study thermodynamics in f ( R ) gravity with the disformal transformation. The transformation applied to the matter Lagrangian has the form of γ μ ν = A ( ϕ , X ) g μ ν + B ( ϕ , X ) ∂ μ ϕ ∂ ν ϕ with the assumption of the Minkowski matter metric γ μ ν = η μ ν , where ϕ is the disformal scalar and X is the corresponding kinetic term of ϕ . We verify the generalized first and second laws of thermodynamics in this disformal type of f ( R ) gravity in the Friedmann-Lemaître-Robertson-Walker (FLRW) universe. In addition, we show that the Hubble parameter contains the disformally induced terms, which define the effectively varying equations of state for matter.