  The adiabatic connection formula of ground-state density functional theory
relates the correlation energy to a coupling-constant integral over a purely
potential contribution, and is widely used to understand and improve
approximations. The corresponding formula for thermal density functional theory
is cast as an integral over temperatures instead, ranging upward from the
system's physical temperature. We also show how to relate different different
correlation components to each other, either in terms of temperature- or
coupling-constant integrations. We illustrate our results on the uniform
electron gas.
