  We delve into the first-order thermodynamics of Horndeski gravity, focusing
on spatially flat, homogeneous, and isotropic cosmologies. Our exploration
begins with a comprehensive review of the effective fluid representation within
viable Horndeski gravity. Notably, we uncover a surprising alignment between
the constitutive relations governing the "Horndeski fluid" and those of
Eckart's thermodynamics. Narrowing our focus, we specialize our discussion to
spatially flat Friedmann-Lema{\^i}tre-Robertson-Walker spacetimes. Within this
specific cosmological framework, we systematically analyze two classes of
theories: shift-symmetric and asymptotically shift-symmetric. These theories
are characterized by a non-vanishing braiding parameter, adding a nuanced
dimension to our investigation. On the one hand, unlike the case of the
"traditional" scalar-tensor gravity, these peculiar subclasses of viable
Horndeski gravity never relax to General Relativity (seen within this formalism
as an equilibrium state at zero temperature), but give rise to additional
equilibrium states with non-vanishing viscosity. On the other hand, this
analysis further confirms previous findings according to which curvature
singularities are "hot" and exhibit a diverging temperature, which suggests
that deviations of scalar-tensor theories from General Relativity become
extreme at spacetime singularities. Furthermore, we provide a novel exact
cosmological solution for an asymptotically shift-symmetric theory as a toy
model for our thermodynamic analysis.
