Thermal channels of scalar and tensor waves in Jordan-frame scalar-tensor gravity

We study first-order scalar and tensor perturbations of Jordan-frame scalar-tensor gravity about a spatially flat Friedmann-Lema\^{}tre-Robertson-Walker (FLRW) background using the Einstein-like effective-fluid decomposition of the scalar sector. In the scalar-gradient frame, we derive the perturbed effective density, pressure, heat flux, and anisotropic stress, and show that they admit an exact Eckart-type constitutive identification at linear order. We then show that these same quantities appear explicitly and exhaustively in the linearized field equations: the scalar Hamiltonian, momentum, trace, and traceless Einstein-like equations are governed, respectively, by the effective density, heat-flux, pressure, and anisotropic-stress channels, while the tensor propagation equation is governed by the transverse-traceless anisotropic-stress channel. In particular, the Jordan-frame modification of gravitational-wave damping is identified with the effective transverse-traceless anisotropic stress of the scalar sector. We also derive the perturbed evolution equation for the invariant product κT, clarify its gauge behavior, and show that flux matching on FLRW fixes only the background value κT, not its perturbation. These results leave open the possibility that gravitational waves in scalar-tensor gravity admit a deeper thermodynamic characterization, perhaps even an intrinsic one, although the present analysis establishes this only at the level of an effective constitutive description.