  Legendre invariant metrics have been introduced in Geometrothermodynamics to
take into account the important fact that the thermodynamic properties of
physical systems do not depend on the choice of thermodynamic potential from a
geometric perspective. In this work, we show that these metrics also have a
statistical origin which can be expressed in terms of the average and variance
of the differential of the microscopic entropy. To show this, we use a
particular reparametrization of the coordinates of the corresponding
thermodynamic phase space.
