  Motivated by recent progresses in the field of massive gravity, the paper at
hand investigates the thermodynamical structure of black holes with three
specific generalizations: i) Gauss-Bonnet gravity which is motivated from
string theory ii) PMI nonlinear electromagnetic field which is motivated from
perspective of the QED correction iii) massive gravity which is motivated by
obtaining the modification of standard general relativity. The exact solutions
of this setup are extracted which are interpreted as black holes. In addition,
thermodynamical quantities of the solutions are calculated and their critical
behavior are studied. It will be shown that although massive and Gauss-Bonnet
gravities are both generalizations in gravitational sector, they show opposing
effects regarding the critical behavior of the black holes. Furthermore, a
periodic effect on number of the phase transition is reported for variation of
the nonlinearity parameter and it will be shown that for super charged black
holes, system is restricted in a manner that prevents it to reach the critical
point and acquires phase transition. In addition, the effects of geometrical
structure on thermodynamical phase transition will be highlighted.
