The Early Paleozoic featured nine intervals of strong expansion of an upper slope, dysoxic/anoxic (d/a) water mass with eustatic rise or epeirogenic transgression. Strong expansion of this d/a water mass led to deposition of time-specific, macroscale alternations of dark grey-black mudstone within oxic, green to red mudstone on the middle–lower slope. This d/a facies even onlapped warm- (carbonate) and cool-water (siliciclastic) shelves. As in the Mesozoic, d/a muds were deposited in shallow water, perhaps tens of metres deep, with sea-level rise. These nine d/a macroscale alternations correspond to intervals of “global hyperwarming”—times of very intense greenhouse conditions that resulted from a feedback initiated by higher insolation and heat storage as shallow seas onlap tropical palaeocontinents. Warm epeiric seas heated the ocean, and thermal expansion accelerated eustatic rise. Ever more extensive epeiric seas heightened oceanic and global temperature as heat storage capacity increased. Deep ocean circulation intensity fell below that of a greenhouse interval and lead to d/a deposition low on the slope and on the platforms to provide the signature of global hyperwarming. Global hyperwarming differs from a hothouse interval as it does not require CO₂ input from large igneous provinces to produce high temperatures and never shows deep-sea anoxia. Late Ordovician and Late Devonian black mudstones that cover much of Laurentia record epeirogenic transgressions that led to global hyperwarming, and suggest that cold water upwelling or plant terrestrialisation had nothing to do with epeiric sea anoxia. Global hyperwarming reduced oxygen solubility in these seas, and erosion of orogens produced muddy water that limited light penetration and promoted shallow-water anoxia. The global hyperwarming hypothesis means that relative eustatic and epeirogenic sea levels complement the effect of global pCO₂ on climate, and sea level must also be regarded as a primary driver of Phanerozoic climate.