Finite temperature density functional theory provides, in principle, an exact description of the thermodynamical equilibrium of many-electron systems. In practical applications, however, the functionals must be approximated. Efficient and physically
meaningful approximations can be developed if relevant properties of the exact functionals are known and taken into consideration as constraints. In this work, derivations of exact properties and scaling relations for the main quantities of finite temperature density functional theory are presented. In particular, a coordinate scaling transformation at finite temperature is introduced and its consequences are elucidated.
