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 $\g_{\m\n} = A(\phi,X)g_{\m\n} + B(\phi,X)\pa_\m\f\pa_\n\f$ with the assumption of the Minkowski matter metric ${\text{g}}_{\text{m}\text{n}}={\text{e}}_{\text{m}\text{n}}$, where $\varphi$ is the disformal scalar and $X$ is the corresponding kinetic term of $\varphi$. 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.