It is well known that contact geometry gives the appropriate framework for formulating thermodynamics: Thermodynamic states can be interpreted as Legendrian submanifolds of a certain contact manifold. The existence of a metric on thermodynamic states has also received some attention in the last decades. The metric can be interpreted as the variance of an underlying probability measure. Less studied is the action of the affine group that appears naturally in this context as the group preserving the variance. We... more