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We have discussed the correspondence of the well-accepted $f(G)$ gravitytheory with two dark energy models : $(m,n)$-type holographic dark energy[$(m,n)$-type HDE] and entropy-corrected $(m,n)$-type holographic dark energy.For this purpose, we have considered the power law form of the scale factor$a(t)=a_{0}t^{p}, p>1$. The reconstructed $f(G)$ in these models have beenfound and the models in both cases are found to be realistic. We have alsodiscussed the classical stability issues in both models. The $(m,n)$-type HDEand its entropy-corrected versions are more stable than the ordinary HDE model.

Reconstruction of $f(G)$ gravity with ordinary and entropy-corrected $(m,n)$ type Holographic dark energy model