  We use the Legendre invariant formalism of geometrothermodynamics to
investigate the geometric properties of the equilibrium space of a spherically
symmetric phantom black hole with electric charge and dilaton. We find that at
certain points of the equilibrium space, the thermodynamic curvature is
characterized by the presence of singularities that are interpreted as phase
transitions. We also investigate the phase transition structure by using the
standard approach of black hole thermodynamics based upon the analysis of the
heat capacity and response functions. We show compatibility between the two
approaches. In addition, a new type of phase transition is found which is due
to the presence of phantom energy and corresponds to a transition between black
hole states with different stability properties.
