Holographic Connection of f(G) Gravity Through Barrow and a Generalized Version of Holographic Dark Fluid

In the study presented in this work, we use the holographic principle taking into account the Barrow entropy rather than the conventional Bekenstein–Hawking one to develop a holographic model for dark energy in the $f\left(G\right)$ gravity. The former results from the attempt to include quantum gravitational effects into the cosmological framework and, in accordance with the gravity–thermodynamic conjecture, into black hole physics. To investigate the cosmological implications of our model for $f\left(G\right)$ modified gravity, we discuss the cosmic implementation of the most generalized type of holographic dark energy, called Nojiri-Odintsov holographic dark energy (NO-HDE), and a particular example of it, called Barrow holographic dark energy. This is accomplished by adding to the $f\left(G\right)$ model a well-known power law form of the scale factor $a\left(t\right)$ and the Holographic dark energy. It is found that the reconstructed $f\left(G\right)$ satisfies a necessary condition for a realistic modified gravity model. Additionally, the reconstruction models are examined in the four energy scenarios. Lastly, the relationship to observational boundaries is examined, and the reconstructed EoS parameter is verified to fall within the constraints determined in the literature by utilizing observational data sets from BAO, SNLS3, and $\text{Planck}+\text{WMAP}9+\text{Wiggle}Z$.