The capacity to extract oxygen from the environment and transport it to respiring tissues in support of metabolic demand reportedly has implications for species' thermal tolerance, body size, diversity and biogeography. Here, we derived a quantifiable linkage between maximum and basal metabolic rate and their oxygen, temperature and size dependencies. We show that, regardless of size or temperature, the physiological capacity for oxygen supply precisely matches the maximum evolved demand at the highest persistently available oxygen pressure and this is the critical PO2 for the maximum metabolic rate, Pcrit-max. For most terrestrial and shallow-living marine species, Pcrit-max is the current atmospheric pressure, 21 kPa. Any reduction in oxygen partial pressure from current values will result in a calculable decrement in maximum metabolic performance. However, oxygen supply capacity has evolved to match demand across temperatures and body sizes and so does not constrain thermal tolerance or cause the well-known reduction in mass-specific metabolic rate with increasing body mass. The critical oxygen pressure for resting metabolic rate, typically viewed as an indicator of hypoxia tolerance, is, instead, simply a rate-specific reflection of the oxygen supply capacity. A compensatory reduction in maintenance metabolic costs in warm-adapted species constrains factorial aerobic scope and the critical PO2 to a similar range, between ∼2 and 6, across each species' natural temperature range. The simple new relationship described here redefines many important physiological concepts and alters their ecological interpretation.