In this post Feynman is quoted stating that GR can potentially remove the inherent ambiguities implied by Poynting's theorem in the definition of EM energy density and energy flux.
Given $u=\frac{1}{2}\epsilon_0 |\mathbf{E}|^2+\frac{1}{2}\mu_0 |\mathbf{H}|^2$ and $\mathbf{S} = \mathbf{E}\times \mathbf{H}$ one has Poynting's theorem: $$\frac{\partial u}{\partial t} + \nabla \cdot \mathbf{S}=0 \tag{1}\label{1}$$ Now for any pair of smooth functions $f(\mathbf{x},t),\mathbf {g}(\mathbf{x},t)$ that also satisfy $\frac{\partial f}{\partial t} + \nabla \cdot \mathbf{g}=0$ we also get $$\frac{\partial u'}{\partial t} + \nabla \cdot \mathbf{S'}=0 \tag{2}\label{2}$$ where $u'=u+f$ and $\mathbf{S'}=\mathbf{S}+\mathbf{g}$.
My question: If this modification is made then what happens to the black body parameters that depend on temperature $T$, namely: $u=aT^4$ and $\sigma = \frac{4}{3}aT^3$ (entropy density), $p=-\psi=\frac{1}{3}u$ (radiation pressure $p$, free energy density $\psi$)? Should we have to make $f,\mathbf {g}$ temperature dependent, too?
