Jupiter and Saturn emit nearly twice the thermal energy they receive from the Sun. Although insolation decreases toward the poles, the large-scale outward heat flux is nearly uniform, with smaller-scale latitudinal undulations that correlate with the zonal jet streams. Here we present numerical models of rapidly rotating, turbulent 3-D convection in geometrically thin, uniformly forced layers of Boussinesq fluid that approximate the deep convection zones of Jupiter and Saturn. In previous studies we have demonstrated that such models generate zonal flows comparable to those observed on the gas giants. By analysing the simulated patterns of convective heat transfer, we show here that deep convection in the gas giants can explain the anomalously uniform large-scale thermal emissions as well as the jet-scale variations. In particular, we find that convective heat transfer by quasi-geostrophic thermal plumes in relatively thin spherical shell geometry generates an outward heat flow pattern with a broad equatorial minimum and peaks at the poles. The results suggest an alternative to the hypothesis that insolation controls the large-scale patterns of heat flux and zonal flow on the gas giants. Instead, we propose that the large-scale thermal and zonal flow fields originate deep within the planets' molecular envelopes. © 2008 The Authors Journal compilation © 2008 RAS.