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We introduce a novel adjoint scheme for topology optimization to enhance the absorption in nanostructures made of lossy dispersive media. The method employs the complex-conjugate pole-residue (CCPR) model to account for the optical dispersion of arbitrary (linear) materials. Its integration within a parallel time-domain Maxwell’s equations solver enables an efficient optimization of dispersive nanostructures over a broad frequency range. Our method is demonstrated by designing Gold and Silicon nanoparticles with enhanced absorption efficiency and a broadband absorbing Silicon metasurface in the visible-ultraviolet regime.