The Three famous Postulates of Thermodynamics.

1. There exists a function called the entropy of extensive parameters of any system, defined for all stabilized states having the property that the entropy maximizes.
2. The entropy of a composite system is additive over constituent subsystems. As a function, it's also continuous, differentiable, and monotically increasing with respect to energy.
3. If (∂u/∂S)_V,N=0, then S=0

The only other thing of note is notice in Postulate Two:

(The Entropy is a) monotonically increasing with respect to energy.

in other words:

∂S/∂U > 0

1/T > 0

Temperature can never be negative.