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We demonstrate a computational inverse design method for optimizing broadband-absorbing metasurfaces made of arbitrary dispersive media. Our figure of merit is the time-averaged instantaneous power dissipation in a single unit cell within a periodic array. Its time-domain formulation allows capturing the response of arbitrary dispersive media over any desired spectral range. Employing the time-domain adjoint method within a topology optimization framework enables the design of complex metasurface structures exhibiting unprecedented broadband absorption.We applied the method to a thin-film Silicon-on-insulator configuration and explored the impact of structural and (time-domain inherent) excitation parameters on performance over the visible–ultraviolet. We provide a physical insight into the dissipation mechanism of the optimized structures. Since our incorporated material model can represent any linear material, the method can also be applied to other all-dielectric, plasmonic, or hybrid configurations.