OBJECTIVE: Modeling and evaluating a series of power law descriptions for boundary conditions of undiminished cognitive capacities under thermal stress. BACKGROUND: Thermal stress degrades cognition, but precisely which components are affected, and to what degree, has yet to be fully determined. With increasing global temperatures, this need is becoming urgent. Power-law distributions have proven their utility in describing differing natural mechanisms, including certain orders of human performance, but never as a rationalization of stress-altered states of attention. METHOD: From a survey of extant empirical data, absolute thresholds for thermal tolerance for varying forms of cognition were identified. These thresholds were then modeled using a rational power-law description. The implications of the veracity of that description were then identified and analyzed. RESULTS: Cognitive performance thresholds under thermal stress are advanced as power-law relationships, t = f(T) = c[(T - Tref)/Tref](-alpha). Coherent scaling parameters for diverse cognitive functionalities are specified that are consistent with increases in deep (core) body temperature. Therefore, scale invariance provides a "universal constant," viz, 20% detriment in mental performance per 10% increase in T deviation, from a comfortable reference temperature Tref. CONCLUSION: We know the thermal range within which humans can survive is quite narrow. The presented power-law descriptions imply that if making correct decisions is critical for our future existence, then our functional thermal limits could be much more restricted than previously thought. APPLICATION: We provide our present findings, such that others can both assess and mitigate the effects of adverse thermal loads on cognition, in whatever human scenario they occur.