I will talk about recent developments in the finite temperature bootstrap of conformal field theories. After reviewing some salient aspects of the general problem, I will focus on the holographic context. Specifically, I will consider weakly-coupled QFT in AdS at finite temperature, and compute the holographic thermal two-point function of scalar operators in the boundary theory. I will present analytic expressions for leading corrections due to local quartic interactions in the bulk, with an arbitrary number of derivatives and for any number of spacetime dimensions. The solutions are fixed by judiciously picking an ansatz and imposing consistency conditions. The conditions include analyticity properties, consistency with the operator product expansion, and the Kubo-Martin-Schwinger condition. For the case without any derivatives, I will demonstrate agreement with an explicit diagrammatic computation. The structure of the answer is suggestive of a thermal Mellin amplitude. Additionally, I will present a simple dispersion relation for thermal two-point functions which reconstructs the function from its discontinuity.