Second-order approximation of Nojiri–Odintsov infrared cutoff through heuristic expansion: A refined framework for holographic dark energy dynamics with Ricci–Gauss–Bonnet form

This work presents a unified treatment of Nojiri–Odintsov holographic dark energy, encompassing its Ricci–Gauss–Bonnet realization and a second-order approximation as a particular case of the generalized framework. By incorporating higher-order curvature corrections from the Gauss–Bonnet invariant and the Ricci scalar, we construct an extended HDE model and study the evolution of the equation-of-state parameter in both interacting and non-interacting scenarios. The analysis reveals possible quintessence-to-phantom transitions, with observational validation from cosmic chronometer and Planck 2018 datasets. The outcomes indicate the role of higher-curvature corrections provide a viable description of late-time cosmic acceleration. Finally, within the Ricci–Gauss–Bonnet and Nojiri–Odintsov holographic frameworks, we find that the reconstructed $w_{\mathrm{tot}}\left(z\right)$ exhibits the DESI-preferred quintessence–phantom crossing, although the Hubble tension between Planck and SH0ES determinations of $H_0$ remains unresolved.