Comment by rotskoff

Statistical physicist here. Negative temperatures occur when a system has a finite number of high energy states. On average, when temperature increases, both the energy and the "randomness" or entropy of a configuration increase as well. Of course, if there are only a few high energy states available, then the randomness will not increase, it will decrease. That's negative temperature! Because we define temperature as the rate of change of the energy with respect to the energy, in systems like the one I described, this rate of change becomes negative.