We formulate a first-order thermodynamic description of Jordan-frame tensor-multiscalar gravity. From the Einstein-like field equations we obtain the exact covariant $1+3$ decomposition of the geometric sector and write it as an effective imperfect fluid. The thermodynamic interpretation is understood in the first-order Eckart sense: the effective temperature, conductivity, entropy current, and entropy production are meaningful only on branches where the geometric dissipative variables satisfy the appropriate matching and integrability conditions. In a generic frame, the heat flux admits the exact decomposition $q_a^{(g)}=-\chi (a_a+W_a)$, with $\chi =-\dot{\mathcal{F}}/(8\pi \mathcal{F})$ and with $W_a$ encoding the residual temperature-gradient sector. In the $\mathcal{F}$-comoving frame this gives the inertial variable ${\chi }_{\mathcal{F}}\equiv K_{\mathcal{F}}{T}_{\mathcal{F}}$ together with a generally nonvanishing spatial contribution $W_a^{(\mathcal{F})}$, sourced by scalar directions not aligned with the coupling. Thus the multifield thermal description is not generically reducible to a single $KT$-type quantity. We derive transport equations for ${\chi }_{\mathcal{F}}$, for the field-space thermal vector ${\chi }^A$ and covector ${\chi }_A$, and for the residual gradient sector. We also introduce the diagnostics ${\mathfrak{D}}_{\chi }={\chi }_A{\chi }^A$ and ${\mathfrak{D}}_{\mathrm{grad}}={\mathcal{B}}_{AB}{\mathrm{D}}_a^{(\mathcal{F})}{\varphi }^A{\mathrm{D}}_{(\mathcal{F})}^a{\varphi }^B$. Their interpretation depends on the kinetic matrix ${\mathcal{B}}_{AB}$: they are canonical contractions when ${\mathcal{B}}_{AB}$ is nondegenerate, but become non-negative normlike diagnostics only when ${\mathcal{B}}_{AB}$ is positive definite; if ${\mathcal{B}}_{AB}$ is degenerate, no model-independent inverse field-space metric or full scalar norm is available without extra structure. With this qualification, these diagnostics show that 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, state the assumptions required for non-negative entropy production, and show that homogeneous cosmology suppresses the spatial sector while retaining nontrivial timelike multiscalar thermal dynamics.

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• Alternative gravity theories
• Fluids & classical fields in curved spacetime
• Nonequilibrium & irreversible thermodynamics