The density functional theory and its extension to ensembles of excited states can be formalized as thermodynamics. However, these theories are not unique because one of their key quantities, the kinetic energy density, can be defined in several ways. Usually, the everywhere positive gradient form is applied; however, other forms are also acceptable, provided they integrate to the true kinetic energy. Recently, a kinetic energy density of the ground-state theory maximizing the information entropy has been proposed. Here, ensemble kinetic energy density, leading to extremum information entropy, is derived via constrained search. The corresponding ensemble temperature is found to be constant.