  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 Nordstr\"om gravity, which contains
only a scalar degree of freedom and has negative temperature relative to
general relativity.
