This study presents a comprehensive first-principles investigation of the optoelectronic and thermoelectric properties of aluminum antimonide (AlSb) in its cubic (F-43m) and hexagonal (P63mc) phases. Structural optimization was performed using the SCAN functional, and all electronic and optical properties were evaluated using the modified Becke-Johnson potential combined with the Hubbard correction (mBJ+U), which best describes the band-edge electronic structure, explicitly accounting for the contribution of the d-states of the Sb half-core, which cannot be adequately accounted for by conventional functionals and may be overestimated by hybrid approaches. Both AlSb phases are found to be quasi-direct bandgap semiconductors, with calculated band gaps of 1.71 eV for the cubic phase and 1.50 eV for the hexagonal phase, in good agreement with available experimental data. The optical response reveals strong absorption in the visible and ultraviolet regions, moderate reflectivity, and high refractive indices, indicating pronounced light-matter interaction characteristic of III-V semiconductors. The hexagonal phase exhibits enhanced low-energy optical absorption due to its reduced symmetry and narrower band gap. Thermoelectric analysis demonstrates large negative Seebeck coefficients, thermally activated carrier generation, and a monotonic increase of the power factor with carrier concentration for both phases. The cubic phase shows higher power factor values due to enhanced carrier mobility, whereas the hexagonal phase benefits from reduced thermal conductivity, which is favorable for thermoelectric performance at elevated temperatures. These results establish AlSb as a multifunctional semiconductor with tunable optoelectronic and thermoelectric properties and highlight the importance of an accurate treatment of Sb d-electron effects for reliable property prediction.