The thermodynamic definition of temperature is simply:
$$T=\left(\frac{\partial{U}}{\partial{S}} \right)_{V,\ n}$$
As you see there is no absolute value involved. This implies that the partial derivative can therefore be negative, and so the system would lose internal energy as it is increasing its entropy. Is this a purely theoretical exercise, or are there systems that behave this way?
In this video1 is explained how through entropy is possible to define a negative temperature to explain how laser light works, and in this video2 is explained that is because of these negative temperatures why laser light could be focus on points that achieve higher temperatures than on the laser pointer emission, thing that will violate the 2nd law of thermodynamics in the case of natural light (due to etendue conservation).
