Tsallis Nonextensive Entropy and Holographic Dark Energy: A Thermodynamic Approach to f(G) Gravity Reconstruction

In this work, we present a comprehensive reconstruction of modified Gauss--Bonnet gravity within the framework of Tsallis nonextensive entropy and holographic dark energy. By employing the Tsallis entropy formalism, a particular case of the most generalized Nojiri--Odintsov entropy formalism, the holographic energy density acquires a generalized nonextensive form, which is then embedded into f(G) gravity to generate a purely geometric description of dark energy. The reconstruction procedure is implemented for three different scale factor choices, namely, hybrid, linearly truncated hybrid, and quadratically truncated hybrid, using the Granda--Oliveros infrared cutoff. The resulting f(G) functions are found to exhibit monotonic growth, indicating negligible deviation from General Relativity in the early universe and significant contributions at late times, thereby supporting accelerated expansion. The reconstructed equation-of-state parameter remains negative throughout the evolution, with possible transitions between phantom and quintessence regimes depending on the model parameters. Further, the adiabatic index exhibits a positive and monotonic evolution, thereby confirming the dynamical stability and thermodynamic consistency of the reconstructed f(G) model. Finally, a detailed thermodynamic investigation confirms that the total entropy variation stays positive, ensuring the validity of the Generalized Second Law of Thermodynamics. Overall, the results establish that the Tsallis holographic dark energy model in f(G) gravity yields a thermodynamically consistent and dynamically viable framework for explaining late-time cosmic acceleration.