A system H with a Hagedorn-like mass spectrum imparts its unique temperature T_H to any other system coupled to it. An H system radiates particles in preexisting physical and chemical equilibrium. These particles form a saturated vapor at temperature T_H. This coexistence describes a first order phase transition. An H system is nearly indifferent to fragmentation into smaller H systems. A lower mass cut-off in the spectrum does not significantly alter the general picture. These properties of the Hagedorn thermostats naturally explain a single value of hadronization temperature observed in elementary particle collisions at high energies and lead to some experimental predictions.
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