We study the response of a starved Kerr black hole magnetosphere to abrupt changes in the intensity of disk emission and in the global magnetospheric current, by means of 1D general relativistic particle-in-cell simulations. Such changes likely arise from the intermittency of the accretion process. We find that in cases where the pair production opacity contributed by the soft disk photons is modest, as in, e.g., M87, such changes can give rise to delayed, strong TeV flares, dominated by curvature emission of particles accelerated in the gap. The flare rise time, and the delay between the external variation and the onset of the flare emitted from the outer gap boundary, are of the order of the light crossing time of the gap. The rapid, large amplitude TeV flares observed in M87 and, perhaps, other AGNs may be produced by such a mechanism.