We report the first observation of the spin Hall conductivity spectrum in GaAs at room temperature. Our terahertz polarimetry with a precision of several μrads resolves the Faraday rotation of terahertz pulses arising from the inverse spin Hall effect of optically injected spin-polarized electrons. The obtained spin Hall conductivity spectrum exhibits an excellent quantitative agreement with theory, demonstrating a crossover in the dominant origin from impurity scattering in the dc regime to the intrinsic Berry-curvature mechanism in the terahertz regime. Our spectroscopic technique opens a new pathway to analyze anomalous transports related to spin, valley, or orbital degrees of freedom.