  We investigate $4D$ Einstein-Gauss-bonnet gravity coupled to exponential
electrodynamics in AdS background and found a static, spherically symmetric
exact black hole solution. The horizon structure of black hole is discussed.
Treating the cosmological constant as pressure, we drive first law of black
hole thermodynamics in extended phase space. We compute the electrostatic
potential and vacuum polarization of the black hole. To study local
thermodynamical stability specific heat of the black hole is computed. Finally,
we study Helmholtz free energy to analyse the global stability of black holes.
