Plugging the Fermi-Dirac distribution into the integral of a Feynman diagram

In this paper https://arxiv.org/pdf/1211.6442.pdf they compute the stress-stress correlation function for a Chern insulator by using a Feynman diagram with a fermi loop and an ingoing and outgoing "graviton" corresponding to the stress tensor (equation 98). What I am wondering is, if I wanted to compute this in the case of constant temperature, would it be possible to just insert a Fermi-Dirac distribution into this integral over momentum space? This sounds pretty handwavy but when I do this then the result I get is completely consistent with another method of computing this response where I start from the Berry curvature and use kinetic theory so I am lead to believe that it is a valid thing to do.