First-order thermodynamics of multi-scalar-tensor gravity

We formulate a first-order thermodynamic description of Jordan-frame tensor--multi-scalar gravity. From the Einstein-like field equations we obtain the exact covariant $1+3$ decomposition of the geometric sector and interpret it as an effective imperfect fluid. In a generic frame, the heat flux can be written exactly as $q_a^{(g)}=-\chi(a_a+W_a)$, with $\chi=-\dot{\FF}/(8\pi\FF)$, where $\FF=\FF(\phi^C)$ is the nonminimal coupling function, and with $W_a$ the residual temperature-gradient sector. In the $\FF$-comoving frame this yields the inertial variable $\chi_{\FF}\equiv K_{\FF}T_{\FF}$ together with a generally nonvanishing spatial contribution $W_a^{(\FF)}$ sourced by scalar directions not aligned with the coupling, showing that the multi-field thermodynamic description is not generically reducible to a single $KT$-type quantity. We derive transport equations for $\chi_{\FF}$, for the field-space thermal vector $\chi^A$ and covector $\chi_A$, and for the residual gradient sector. We further introduce the scalar diagnostics $\mathfrak D_\chi=\chi_A\chi^A$ and $\mathfrak D_{\rm grad}=\mathcal{B}_{AB}\D_a^{(\FF)}\phi^A\D^a_{(\FF)}\phi^B$, where $\mathcal{B}_{AB}$ is the field-space kinetic matrix of the multi-scalar theory and $\D_a^{(\FF)}$ is the covariant derivative projected orthogonally to the $\FF$-comoving 4-velocity. These diagnostics characterize the full time-like and spatial multi-scalar sectors and lead to a precise GR-attractor criterion: freezing the effective coupling is, in general, weaker than full relaxation to the GR sector. Finally, we construct the entropy current and entropy production in the coupling frame and show that homogeneous cosmology suppresses the spatial sector while retaining nontrivial time-like multi-scalar thermal dynamics.