  We studied the thermodynamics of the black holes with Gauss-Bonnet correction
term in $3+1-$ dimensional AdS spacetime. It is known that the term has no
effect on the equation of motion, however, it modifies the entropy formula of
Wald as it has an effect of the Gauss-Bonnet parameter term. We studied charged
black holes, namely, Reissner-Nordstr\"{o}m and Born-Infeld under this regime.
We encountered divergences in heat capacity. After eliminating the possibility
of first order phase transition, we applied two well trusted methods from
standard thermodynamics, namely, Ehrenfest scheme and Ruppeiner's state space
geometry analysis to ensure the second order nature of phase transition points.
Effects of Gauss-Bonnet and Born-Infeld parameter are also discussed.
