  We consider the two classes cosh and sinh of normal and phantom black holes
of Einstein-Maxwell-dilaton theory. The thermodynamics of these holes is
characterized by heat capacities that may have both signs depending on the
parameters of the theory. Leaving aside the normal Reissner-Nordstr\"om black
hole, it is shown that only some phantom black holes of both classes exhibit
critical phenomena. The two classes share a nonextremality, but special,
critical point where the transition is continuous and the heat capacity, at
constant charge, changes sign with an infinite discontinuity. This point yields
a classification scheme for critical points. It is concluded that the two
unstable and stable phases coexist on one side of the criticality state and
disappear on the other side, that is, there is no configuration where only one
phase exists. The sinh class has an extremality critical point where the
entropy diverges. The transition from extremality to nonextremality with the
charge held constant is accompanied by a loss of mass and an increase in the
temperature. A special case of this transition is when the hole is isolated
(microcanonical ensemble), it will evolve by emission of energy, which results
in a decrease of its mass, to the final state of minimum mass and vanishing
heat capacity. The Ehrenfest scheme of classification is inaccurate in this
case but the generalized one due to Hilfer leads to conclude that the
transition is of order less than unity. Fluctuations near criticality are also
investigated.
