Maximal temperature of the gas in AdS space-time

Assuming only statistical mechanics and general relativity, we calculate the maximal temperature of gas of particles placed in AdS space-time. If two particles with a given center of mass energy come close enough, according to classical gravity they will form a black hole. We focus only on the black holes with Hawking temperature lower than the environment, because they do not disappear. The number density of such black holes grows with the temperature in the system. At a certain finite temperature, the thermodynamical system will be dominated by black holes. This critical temperature is lower than the Planck temperature for the values of the AdS vacuum energy density below the Planck density. This result might be interesting from the AdS/CFT correspondence point of view, since it is different from the Hawking-Page phase transition, and it is not immediately clear what effect dynamically limits the maximal temperature of the thermal state on the CFT side of the correspondence.