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In the context of the recently proposed first-order thermodynamics ofscalar-tensor gravity, we discuss the possibility of zero-temperature states ofequilibrium other than Einstein gravity, including pathological Brans-Dicketheory, Palatini $f(R)$ gravity, and cuscuton gravity, all with non-dynamicalscalar fields. The formalism is extended to Nordstr\"om gravity, which containsonly a scalar degree of freedom and has negative temperature relative togeneral relativity.

Peculiar thermal states in the first-order thermodynamics of gravity