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In this paper, we discuss cosmological application of holographic Dark Energy(HDE) in the framework of $f(G)$ modified gravity. For this purpose, weconstruct $f(G)$ model with the inclusion of HDE and a well-known power lawform of the scale factor $a(t)$. The reconstructed $f(G)$ is found to satisfy asufficient condition for a realistic modified gravity model. We findquintessence behavior of effective equation of state (EoS) parameter$\omega_{DE}$ through energy conditions in this context. Also, we observe thatthe squared speed of sound $v_s^2$ remains negative which shows the instabilityof HDE $f(G)$ model.

Correspondence between $f(G)$ Gravity and Holographic Dark Energy via Power-law Solution