  Universal exact conditions guided the construction of most ground-state
density functional approximations in use today. We derive the relation between
the entropy and Mermin free energy density functionals for thermal density
functional theory. Both the entropy and sum of kinetic and electron-electron
repulsion functionals are shown to be monotonically increasing with
temperature, while the Mermin functional is concave downwards. Analogous
relations are found for both exchange and correlation. The importance of these
conditions is illustrated in two extremes: the Hubbard dimer and the uniform
gas.
