A detailed analysis of how intermittency (i.e., the alternation of dry and rainy periods) modulates the rate at which sub-daily rainfall extremes depend on temperature is presented. Results show that hourly extremes tend to be predominantly controlled by peak intensity, increasing at a rate of approximately 7% per degree in agreement with the Clausius-Clapeyron equation. However, a rapid increase in intermittency upward of 20-25°C is shown to produce local deviations from this theoretical scaling, resulting in lower scaling rates. On the other hand, rapidly decreasing intermittency with temperature between 10 and 20° can result in higher net scaling rates than expected, potentially exceeding Clausius-Clapeyron. In general, the importance of intermittency in controlling the scaling rates of precipitation with temperature grows as we progress from hourly to daily aggregation timescales and beyond. Thermodynamic effects still play an important role in controlling the maximum water-holding capacity of the atmosphere and therefore peak rainfall intensity, but the observational evidence shows that, beyond a few hours, storm totals become increasingly dominated by dynamical factors. The conclusion is that Clausius-Clapeyron scaling alone cannot be used to reliably predict the net effective changes in rainfall extremes with temperature beyond a few hours. A more general scaling model that takes into account simultaneous changes in intermittency and peak intensity with temperature is proposed to help better disentangle these two phenomena (e.g., peak intensity and intermittency). The new model is applied to a large number of high-resolution rain gauge time series in the United States, and results show that it greatly improves the representation of rainfall extremes with temperature, producing a much more consistent and reliable picture of extremes across scales than using Clausius-Clapeyron only.