The upper atmospheres of the four Solar System giant planets exhibit high temperatures that cannot be explained by the absorption of sunlight. In the case of Saturn the temperatures predicted by models of solar heating are approximately 200 K, compared to temperatures of approximately 400 K observed independently in the polar regions and at 30 degrees latitude. This unexplained 'energy crisis' represents a major gap in our understanding of these planets' atmospheres. An important candidate for the source of the missing energy is the magnetosphere, which injects energy mostly in the polar regions of the planet. This polar energy input is believed to be sufficient to explain the observed temperatures, provided that it is efficiently redistributed globally by winds, a process that is not well understood. Here we show, using a numerical model, that the net effect of the winds driven by the polar energy inputs is not to heat but to cool the low-latitude thermosphere. This surprising result allows us to rule out known polar energy inputs as the solution to the energy crisis at Saturn. There is either an unknown--and large--source of polar energy, or, more probably, some other process heats low latitudes directly.