Note that an isolated system with a bounded energy spectrum, such as a two-state quantum system, can have a temperature greater than this. In fact, such a system can have a
negative temperature, which is actually "hotter" than any positive temperature and indicates
population inversion. In such a context, the greatest possible temperature is
negative zero, which means the system is in the state of highest possible energy. For systems with unbounded energy, however, negative temperature states are impossible, and the greatest possible temperature would be positive infinity, if not for the finite limit of "absolute hot" discussed here.