  We establish a comprehensive theoretical framework for systems subjected to a
static uniform temperature gradient, employing the non-equilibrium
Keldysh-Dyson formalism. This framework interprets the statistical force due to
the temperature gradient as a mechanical force, utilizing both Luttinger's
scalar and Moreno-Coleman-Tatara's vector potentials, which collectively
emulate the gauge invariance stemming from the conservation of energy. Our
approach has the ability to treat heat current and heat magnetization on an
equal footing, thereby extending and generalizing previous formalisms. The
derived result for the thermal conductivity is applied to investigate the
thermal characteristics of Weyl magnons in a stacked honeycomb ferromagnet
featuring a trivial insulator phase, a magnon Chern insulator phase, and three
Weyl magnon phases. Against the expectation from the Berry curvature, the
magnon Chern insulator phase exhibits the highest transverse thermal
conductivity.
