  We study some aspects of conformal field theories at finite temperature in
momentum space. We provide a formula for the Fourier transform of a thermal
conformal block and study its analytic properties. In particular we show that
the Fourier transform vanishes when the conformal dimension and spin are those
of a "double twist" operator $\Delta = 2\Delta_\phi + \ell + 2n$. By
analytically continuing to Lorentzian signature we show that the spectral
density at high spatial momenta has support on the spectrum condition $|\omega|
> |k|$. This leads to a series of sum rules. Finally, we explicitly match the
thermal block expansion with the momentum space Green's function at finite
temperature in several examples.
