  Modern cosmological models are constructed in the framework of thermodynamic
approaches developed within a Van der Waals-Maxwell theory of the first-order
phase transitions. In the present work we study a geometrothermodynamics of
two-dimensional first-order phase transition with the distribution of
relaxation times in a configuration space which describes a spacetime with
Newman-Unti-Tamburino-like metric. We utilized the geometrothermodynamical
approach to construct the model of a charged generalized-NUT black hole. We
reveal following features of the black-hole phase transition: there are series
of bifurcations of pitchfork type in dependences of the Gibbs free energy on
the Hawking temperature, and although a scalar Berwald curvature of space
changes sign in the phase transition, black-hole stability depends on sign of
the curvature after the transition.
