Single-object imaging and spectroscopy on telescopes with apertures ranging from ~4 m to 40 m have the potential to greatly enhance the cosmological constraints that can be obtained from LSST. Two major cosmological probes will benefit greatly from LSST follow-up: accurate spectrophotometry for nearby and distant Type Ia supernovae will expand the cosmological distance lever arm by unlocking the constraining power of high-z supernovae; and cosmology with time delays of strongly-lensed supernovae and quasars will require additional high-cadence imaging to supplement LSST, adaptive optics imaging or spectroscopy for accurate lens and source positions, and IFU or slit spectroscopy to measure detailed properties of lens systems. We highlight the scientific impact of these two science drivers, and discuss how additional resources will benefit them. For both science cases, LSST will deliver a large sample of objects over both the wide and deep fields in the LSST survey, but additional data to characterize both individual systems and overall systematics will be key to ensuring robust cosmological inference to high redshifts. Community access to large amounts of natural-seeing imaging on ~2-4 m telescopes, adaptive optics imaging and spectroscopy on 8-40 m telescopes, and high-throughput single-target spectroscopy on 4-40 m telescopes will be necessary for LSST time domain cosmology to reach its full potential. In two companion white papers we present the additional gains for LSST cosmology that will come from deep and from wide-field multi-object spectroscopy.
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