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 $g_{m } = A(phi,X)g_{m } + B(phi,X)pa_mfpa_ f$ with the assumption of the Minkowski matter metric $g_{m } = e_{m }$, where $phi$ is the disformal scalar and $X$ is the corresponding kinetic term of $phi$. We verify the generalized first and second laws of thermodynamics in this disformal type of $f(R)$ gravity in the Friedmann-Lema^{i}tre-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.