  We explore alternative formulations of the analogy between viable Horndeski
gravity and Eckart's first-order thermodynamics. We single out a class of
identifications for the effective stress-energy tensor of the scalar field
fluid that, upon performing the imperfect fluid decomposition, yields
constitutive relations that can be mapped onto Eckart's theory. We then
investigate how different couplings to Einstein's gravity, at the level of the
field equations, can affect the thermodynamic formalism overall. Lastly, we
specialise the discussion to the case of ``traditional'' scalar-tensor theories
and identify a specific choice of the coupling function that leads to a
significant simplification of the formalism.
