This paper uses the Friedmann–Robertson–Walker (FRW) line element to simulate the development of the universe and explores the cosmological implications of Tsallis Holographic Dark Energy (THDE) in the modified $f(T)$ teleparallel gravity framework. Teleparallel gravity, in which torsion substitutes curvature as the geometric structure of spacetime, describes the accelerated expansion of the universe in the absence of exotic matter. The work examines key cosmological parameters, such as the Hubble parameter, equation of state (EOS), deceleration parameter, jerk parameter, and $Om$ diagnostics, under both interacting and non-interacting conditions by integrating the THDE model, derived from non-additive Tsallis entropy, with $f(T)$ gravity. To provide a smooth transition from deceleration to acceleration, the study uses a hyperbolic scale factor, $a(t) = \left [\sinh (\alpha t)\right ]^{\frac{1}{\beta }}$ . It explains how THDE fits into the model of the dynamics of the universe, with a focus on stability analysis using square sound speed and scalar field correspondence. We show that while the non-interacting THDE model remains stable (with $v_{s}^{2}>0$ ), the interacting model can develop instabilities for larger coupling values. Our analysis, calibrated against 57 Hubble data points, yields best-fit parameters $\alpha = {-59.416}_{-0.625}^{+0.625}$ and $\beta = {-1.303}_{-0.022}^{+0.022}$ , with an $R^{2}$ value of 0.9321 and a best-fit $H_{0}$ of 67.8 km/s/Mpc. The study also examines energy conditions, highlighting the role of interaction terms in the development of dark energy (DE) and matter components. Graphical and numerical investigations demonstrate the capacity of the model to reflect empirical data, providing a complete framework for understanding the interactions of DE and teleparallel gravity. This methodology expands the theoretical framework for examining the accelerating expansion of the universe and offers alternate viewpoints on the DE paradigm.
