Although interannual streamflow variability is primarily a result of precipitation variability, temperature also plays a role. The relative weakness of the temperature effect at the annual time scale hinders understanding, but may belie substantial importance on climatic time scales. Here we develop and evaluate a simple theory relating variations of streamflow and evapotranspiration (E) to those of precipitation (P) and temperature. The theory is based on extensions of the Budyko water‐balance hypothesis, the Priestley‐Taylor theory for potential evapotranspiration ( ), and a linear model of interannual basin storage. The theory implies that the temperature affects streamflow by modifying evapotranspiration through a Clausius‐Clapeyron‐like relation and through the sensitivity of net radiation to temperature. We apply and test (1) a previously introduced “strong” extension of the Budyko hypothesis, which requires that the function linking temporal variations of the evapotranspiration ratio (E/P) and the index of dryness ( /P) at an annual time scale is identical to that linking interbasin variations of the corresponding long‐term means, and (2) a “weak” extension, which requires only that the annual evapotranspiration ratio depends uniquely on the annual index of dryness, and that the form of that dependence need not be known a priori nor be identical across basins. In application of the weak extension, the readily observed sensitivity of streamflow to precipitation contains crucial information about the sensitivity to potential evapotranspiration and, thence, to temperature. Implementation of the strong extension is problematic, whereas the weak extension appears to capture essential controls of the temperature effect efficiently.