The heat heat energy must go to new degrees of freedom which open up as the temperature slowly decreases. This requires new physics since in standard physics one cannot imagine this kind of new degrees of freedom with high heat capacity near some limiting temperature.
The notion of Hagedorn temperature TH as this kind of limiting temperature was originally introduced in string theory. The feeded energy goes to opening up degrees of freedom of a vibrating string. The heat capacity of the combined system increases and temperature rise slows down and one approaches TH.
This happens always in the presence of extended objects with an infinite number of degrees of freedom. In TGD the strings are replaced by monopole flux tubes representing new physics and there is an entire hierarchy of Hagedorn temperatures corresponding to the spectrum of string tensions predicted by p-adic length scale hypothesis - new physics again. In living matter physiological temperature could be Hagedorn temperature. In nuclear fusion reactors, magnetic monopole flux tube structures eating the feeded energy could be formed.
For the notion of Hagedorn temperature in astrophysics, see this .
For the notion of TH in biology and physics of life, see this and this . The idea is that the temperature at the magnetic body containing quantum coherent dark matter as phases of the ordinary matter with large value of Planck constant, which controls the biological body, slowly approaches TH, the entropy increases and the biocontrol by MB starts to fai. This would give rise to aging.
For a summary of earlier postings see Latest progress in TGD.