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 ΛCDM model at late times.