The Hagedorn temperature exists because the amount of energy available is high enough that matter particle (quark–antiquark) pairs can be spontaneously pulled from vacuum.
Thus, naively considered, a system at Hagedorn temperature can accommodate as much energy as one can put in, because the formed quarks provide new degrees of freedom, and thus the Hagedorn temperature would be an impassable absolute hot.
This quark matter may have been detected in heavy-ion collisions at SPS and LHC in CERN (France and Switzerland)[3] and at RHIC in Brookhaven National Laboratory (USA).
The states that cause the divergence are spatially big, since the electrons are very far from the protons.
The concept of exponential growth in the number of states was originally proposed in the context of condensed matter physics.
It was incorporated into high-energy physics in the early 1970s by Steven Frautschi and Hagedorn.
It is controlled by the size of the string tension, which is smaller than the Planck scale by some power of the coupling constant.