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Peculiar thermal states in the first-order thermodynamics of gravity

In the context of the recently proposed first-order thermodynamics of scalar-tensor gravity, we discuss the possibility of zero-temperature states of equilibrium other than Einstein gravity, including pathological Brans-Dicke theory, Palatini $f(R)$ gravity, and cuscuton gravity, all with non-dynamical scalar fields. The formalism is extended to Nordström gravity, which contains only a scalar degree of freedom and has negative temperature relative to general relativity.