The density functional theory of atomic electrons in strong magnetic fields is generalized to finite-temperature systems. General integral formulations are developed in the format of Mermin-Kohn Sham finite-temperature density functional theory. The lowest order of the general theory leads to a temperature-dependent extended Thomas-Fermi (TETF)-like functional, which is simple enough to be analyzed. The general theory provides a new way of calculating the equilibrium properties of many-electron systems in strong magnetic fields.