The Cosmological Dynamics of Tsallis Holographic Dark Energy in Saez-Ballester Gravity

We investigate interacting and non-interacting Tsallis Holographic Dark Energy (THDE) models within the framework of Sáez–Ballester (SB) scalar–tensor gravity for anisotropic Bianchi type(BT) II, VIII, and IX Universes. Employing the Hubble horizon as the infrared cutoff, we examine the models without assuming a particular scale factor law. The analysis covers key cosmological parameters, including the deceleration parameter, Hubble parameter, energy densities, skewness, the Equation of State (EoS), and the Squared sound speed. Our findings indicate a continuous phantom-like acceleration (w < −1) with transition redshift zt ≈ 0.67 and negligible late-time anisotropy, consistent with cosmic microwave background (CMB) bounds. Compared to ΛCDM, the THDE models predict an earlier onset of acceleration and a more negative present-day EoS. However, the presence of negative squared sound speed at higher redshifts signals a classical instability of the dark energy fluid. These results highlight THDE as a viable alternative to ΛCDM in anisotropic cosmologies, while motivating further work with alternative cutoffs or stabilising mechanisms to overcome the instability issue.