A careful revision of the rudiments of statistical physics shows that negative temperatures are artefacts of Boltzmann's approximate definition of entropy. Gibbs' version, however, forbids negative absolute temperatures and is consistent with thermodynamics.
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Sokolov, I. Not hotter than hot. Nature Phys 10, 7–8 (2014). https://doi.org/10.1038/nphys2831
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