I don't quite understand. Heat is a measure of the vibrational kinetic energy any individual molecule possesses as it travels along, right? So, shouldn't the maximum temperature just be whatever the unit conversion is for one C-per-oscillation? That is, at no time should a particle be able to be pushed by its vibration into a velocity greater than light, correct?
The kinetic energy goes to infinity as the speed of a particle approaches the speed of light, so there's no upper bound on the kinetic energy a particle can have in special relativity. The formula you possibly had in mind, mv^2/2, is valid only for v much less than c; for relativistic velocities the energy of a particle is equal to mc^2/sqrt(1 - v^2/c^2).
BTW, it's any kinetic energy that contributes to the temperature, not necessarily vibrational.
