Given a stationary state for a noncommutative flow, we study a boundedness condition, depending on a positive parameter beta, which is weaker than the KMS equilibrium condition at inverse temperature beta. This condition is equivalent to a holomorphic property closely related to the one recently considered by Ruelle and D'Antoni-Zsido and shared by a natural class of non-equilibrium steady states. Our holomorphic property is stronger than the Ruelle's one and thus selects a restricted class of non-equilibrium steady states. We also introduce the complete boundedness condition and show this notion to be equivalent to the Pusz-Woronowicz complete passivity property, hence to the KMS condition. In Quantum Field Theory, the beta-boundedness condition can be interpreted as the property that localized state vectors have energy density levels increasing beta-subexponentially, a property which is similar in the form and weaker in the spirit than the modular compactness-nuclearity condition. In particular, for a Poincare' covariant net of C*-algebras on the Minkowski spacetime, the beta-boundedness property, for beta greater equal than $2\pi$, for the boosts is shown to be equivalent to the Bisognano-Wichmann property. The Hawking temperature is thus minimal for a thermodynamical system in the background of a Rindler black hole within the class of beta-holomorphic states. More generally, concerning the Killing evolution associated with a class of stationary quantum black holes, we characterize KMS thermal equilibrium states at Hawking temperature in terms of the boundedness property and the existence of a translation symmetry on the horizon.