Perhaps more fundamentally than temperature, the 'hotness' or 'coldness' of a system can be described by the thermodynamic beta instead, which just like temperature is determined by the rate of change of entropy with respect to energy (at constant volume and particle number):
$$betaequivfrac{1}{k_text{B}}left.frac{partial S}{partial E}right|_{V,N}!!=frac{1}{k_text{B}T}text{.}$$
Intuitively, the thermodynamic beta measures the coldness of a system from $-infty$ to $+infty$. Since temperature per se is just the reciprocal of $beta$, up to a factor of Boltzmann's constant $k_text{B}$, that means the coldest is at $Tto 0^+$, approaching zero from positive temperature side, while the hottest is $Tto 0^-$, from the negative. Treating $+0,mathrm{K}$ as different from $-0,mathrm{K}$ may seem bizarre, but that's what we get for taking a reciprocal.
Negative temperatures ($beta<0$) are hotter than any positive temperature ($beta > 0$). They can happen when a system has more high-energy states occupied than low-energy states, such as during population inversion of any laser, or during similar astrophysical phenomena.
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