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 to acquire phase transition. In addition, the effects of geometrical structure on thermodynamical phase transition will be highlighted.
