The widespread correlation between snowline elevation and mountain height is evidence that glacial buzzcutting puts a cap on mountain growth. The match is strongest for mid-latitude ranges, where glacial erosion has persisted over Pleistocene climate cycles and tends to truncate mountain range elevation near the upper limit of the late-Pleistocene snowline. Signs of a glacial buzzsaw are weakest in tropical ranges, where glacial erosion features are sparse and generally restricted to cold periods such as the Last Glacial Maximum (LGM). Here we show that glacial erosion does indeed truncate tropical mountains, often close to the cold-phase snowline. It does so on a cyclic basis, with glacial landscapes expanding during cold periods, and contracting during largely ice-free warm periods as fluvially-driven escarpments encroach on all sides. We find evidence of this cyclicity in the perched terrain of the Chirripó massif in Costa Rica, where surface-exposure age dating and topographic analysis show that LGM denudation occurred across a glacial landscape that has shrunk during post-LGM scarp encroachment. We find a similar story in the Central Range of Taiwan, where scarp encroachment is even more severe. We deduce that, during the Pleistocene, cold-phase glacial erosion has imposed a ceiling on tropical mountain growth, and that even the archetypally steady-state landscape of Taiwan has been subject to strongly cyclic changes in erosion rate.