Quantum Entropy-Driven Modifications to Holographic Dark Energy in $f(G,T)$ Gravity

We present a $f(G,T)$ gravity-based reconstruction of Barrow Holographic Dark Energy (BHDE). This approach extends the conventional HDE model by replacing the Bekenstein-Hawking entropy with Barrow entropy, which encapsulates quantum gravitational corrections to the geometry of black hole horizons. We explore the cosmological dynamics of a flat FRW background filled with a pressureless dust fluid, considering both conserved and non-conserved energy-momentum tensor models. To this end, we employ the Hubble horizon as the infrared cutoff and adopt a power-law ansatz for the scale factor. We then investigate the evolution of key cosmological parameters, including the equation-of-state parameter \( \omega_{GT} \), the deceleration parameter \( q \), and the squared sound speed $v_s^2$. Furthermore, we explore the dynamical behavior in the $\omega_{GT}-\omega'_{GT}$ space. In the case of conserved energy-momentum tensor, our findings indicate that the BHDE model evolves from a quintessence-like regime into the phantom domain. This transition supports the current accelerated expansion of the Universe and offers an improvement over the original HDE model, which does not adequately account for the observed phenomenology. The corresponding $\omega_{GT}-\omega'_{GT}$ trajectory lies within the freezing region. On the other hand, within the non-conserved framework, the BHDE model exhibits phantom-like behavior in the early Universe, subsequently evolving toward either a cosmological constant-like state or a quintessence-like regime. Notably, unlike the conserved scenario, the squared sound speed $v_s^2$ asymptotically attains positive values in the far future. Moreover, the trajectory in the $\omega_{GT}-\omega'_{GT}$ phase space displays a thawing behavior. Finally, we evaluate the observational viability of our results and compare them with predictions from alternative reconstructed DE models.