Structure of the graviton self-energy at finite temperature

We study the graviton self-energy function in a general gauge, using a hard thermal loop expansion which includes terms proportional to $T^4,$ $T^2$ and $\log \mathrm{}\left(T\right).$ We verify explicitly the gauge independence of the leading $T^4$ term and obtain a compact expression for the subleading $T^2$ contribution. It is shown that the logarithmic term has the same structure as the ultraviolet pole part of the $T=0\mathrm{}$ self-energy function. We argue that the gauge-dependent part of the $T^2$ contribution is effectively canceled in the dispersion relations of the graviton plasma, and present the solutions of these equations.