  Motivated by gauge/gravity group in the low energy effective theory of the
heterotic string theory and novel aspects of massive gravity in the context of
lattice physics, the minimal coupling of Gauss-Bonnet-massive gravity with
Born-Infeld electrodynamics is considered. At first, the metric function is
calculated and then the geometrical properties of the solutions are
investigated. It is found that there is an essential singularity at the origin
and the intrinsic curvature is regular elsewhere. In addition, the effects of
massive parameters are studied and black hole solutions with multi horizons are
found in this gravity. Also the conserved and thermodynamic quantities are
calculated, and it is shown that the solutions satisfy the first law of
thermodynamics. Furthermore, using heat capacity of these black holes, thermal
stability and phase transitions are investigated. The variation of different
parameters and related modifications on the (number of) phase transition are
examined. Next, the critical behavior of the Gauss-Bonnet-Born-Infeld-massive
black holes in the context of extended phase space is studied. It is shown that
how the variation of the different parameters affects the existence and absence
of phase transition. Also, it is found that for specific values of different
parameters, these black holes may enjoy the existence of new type of phase
transition which to our knowledge was not observed in black hole physics
before.
