Warm objects actually do weigh more than their counterfactual cold versions haha. The stress energy tensor is the quantity to look at here.
Abstract: "According to the weak form of the equivalence principle all objects fall at the same rate in a gravitational field. However, recent calculations in finite-temperature quantum field theory have revealed that at T>0 heavier and/or colder objects actually fall faster than their lighter and/or warmer counterparts. This unexpected result is demonstrated using elementary quantum mechanical arguments."
Downloadable here: https://www.academia.edu/download/109363694/download.pdf
https://herebeanswers.com/things-weigh-heavier-or-lighter-wh...
why did you choose that one? serious question, because I'm trying to understand your process (and yes, maybe gently challenging it, but unsure if I should, bc you are clearly knowledgeable in specific ways about this)