Coherent perfect absorption (CPA), or anti-lasing, has been so far inherently restricted to continuous wave scenarios, drastically restricting its applications to standard linear steady-state systems. However, future technologies based on enhanced light-matter interactions typically require the dynamic emission and absorption of pulses, as in ultrafast optics, frequency-comb technologies, or spiking neuromorphic networks. Here, we propose to extend the reach of anti-lasing to pulsed operation. We unveil the phenomenon of fast temporal anti-lasing, in which perfect absorption of photons occurs transiently over ultrashort time scales, creating fast absorption pulses associated with broadband absorption frequency combs. This is obtained by leveraging robust topological transitions occurring in a hysteretic scattering system, which is temporally modulated to loop near a CPA singularity. Our work evidences the interplay between intrinsic memory and topology in wave scattering, unveiling the rich physics of Floquet engineering through topological switching. We envision applications in spiking photonic networks with robust emission, routing and detection of spikes, which may form the basis for future analog neuromorphic hardware.