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In this paper we have considered the thermodynamics of a Born-Infeld $AdS$black hole using inputs from the dual boundary field theory. Here, we havevaried the cosmological constant $\Lambda$ and the Newton's gravitationalconstant $G$ along with the Born-Infeld parameter $b$ in the bulk. A noveluniversal critical behaviour of the central charge (occurring in the boundaryconformal field theory) in extended black hole thermodynamics for charged blackholes has been recently observed \cite{mann1}, and we have extended this studyto Born-Infeld $AdS$ black holes. The Born-Infeld parameter has the dimensionof inverse length, therefore, when considered in the first law ofthermodynamics of the bulk in the mixed form which includes the central chargeof the boundary conformal field theory, it modifies the thermodynamic volumeand the chemical potential (which are conjugate to pressure and central chargerespectively). We observe that due to this inclusion of the Born-Infeldnon-linearity in this analysis, the universal nature of the critical value ofthe central charge observed in \cite{mann1} breaks down. We also observe aninteresting behaviour of the free energy of the black hole with Hawkingtemperature due to the variations in both the central charge and theBorn-Infeld parameter. It is also observed in our analysis that for asufficiently small value of the Born-Infeld parameter (small value of thisparameter has more prominent non-linear effects), there exists a critical valueof the temperature below which no black hole can exist.

Phase transition structure and breaking of universal nature of central charge criticality in a Born-Infeld $AdS$ black hole