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We studied the thermodynamics of the black holes with Gauss-Bonnet correctionterm in $3+1-$ dimensional AdS spacetime. It is known that the term has noeffect on the equation of motion, however, it modifies the entropy formula ofWald as it has an effect of the Gauss-Bonnet parameter term. We studied chargedblack holes, namely, Reissner-Nordstr\"{o}m and Born-Infeld under this regime.We encountered divergences in heat capacity. After eliminating the possibilityof first order phase transition, we applied two well trusted methods fromstandard thermodynamics, namely, Ehrenfest scheme and Ruppeiner's state spacegeometry analysis to ensure the second order nature of phase transition points.Effects of Gauss-Bonnet and Born-Infeld parameter are also discussed.

Effects of Gauss-Bonnet gravity on thermodynamics of (3+1)$-D$ AdS black holes