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