Can a Hagedorn system have a temperature other than T_C or can a thermostat have a temperature other than its own?

This is a note intended to complement our paper (nucl-th/0504010) and addressed to the attention of QGP workers interested in bag models, Hagedorn spectra, and the like. It tries to show that with a Hagedorn-like experimental spectrum the partition function can not be calculated and that a canonical description derived for the microcanonical ensemble exists only for a single, fixed temperature.