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 γμν=A(ϕ,X)gμν+B(ϕ,X)∂μϕ∂νϕ with the assumption of the Minkowski matter metric γμν=ημν, 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.