  In this work, the Renyi holographic dark energy (RHDE)and its behaviour has
been explored with the anisotropic and spatially homogeneous Bianchi type-I
Universe in the framework of $f(G)$ gravity. We use IR cutoff as the Hubble and
Granda-Oliveros (GO) horizons. To find a consistent solutions of the field
equations of the models, it is assumed that the deceleration parameter is
defined in terms of function of Hubble parameter $H$. With reference to current
cosmological data, the behaviors of the cosmological parameters relating to the
dark energy model are evaluated and their physical significance is examined. It
is observed that for both the models, the equation of state parameter
approaches to $-1$ at late times. However, the RHDE model with the Hubble
horizon exhibits stability from the squared sound speed, but the RHDE model
with the GO horizon exhibits instability. In both the models, deceleration
parameter and statefinder diagnostic confirm the accelerated expansion of the
Universe and also correspond to the $\Lambda$CDM model at late times.
