  In this work we study thermodynamics of generalized Ayon-Beato and Garcia
(ABG) black hole metric which contains three parameters named as mass $m$,
magnetic charge $q$ and dimensionless coupling constant of nonlinear
electrodynamics interacting field $\gamma$. We showed that central regions of
this black hole behaves as dS(AdS) vacuum space by setting $q<2m (q>2m)$ and in
the case $q=2m$ reaches to a flat Minkowski space. In the large distances this
black hole behaves as a Reissner-Nordstrom BH. However important role of the
charge q is appeared in produce of a formal variable cosmological parameter
which will support pressure coordinate in the thermodynamic perspective of this
black hole in our setup. We should be point that this formal variable
cosmological parameter is different with cosmological constant which comes from
AdS/CFT correspondence and it is effective at large distances as AdS space
pressure. In our setup the assumed pressure is originated from internal
material of the black hole say q and m here. By calculating the Hawking
temperature of this black hole we obtain equation of state. Then we plotted
isothermal P-v curves and heat capacity at constant pressure. They show that
the system participates in the small to large phase transition of the black
hole or the Hawking-Page phase transition which is similar to the Van der Waals
phase transition in the ordinary thermodynamics systems . In fact in the
Hawking-Page phase transition disequilibrium evaporating generalized ABG black
hole reaches to a vacuum AdS space finally.
