Density functional theory (DFT) has become a standard method for ab initio calculations of material properties. However, it has a number of shortcomings, particularly in predicting key properties, such as band gap and optical spectra, which are dependent on excited states. To treat such properties, more accurate approaches such as GW or DFT with hybrid functionals (including HSE, PBE0, and B3LYP, to name a few) can be employed; however, these approaches are unfeasible for many large and/or complex systems due to their high computational cost and large memory requirements. In this work, we investigate the ability to train neural networks of the traditional DFT charge density computed with a standard PBE functional to accurately predict HSE band gaps. We show that a single network PBE charge density functional can predict the HSE band gap of seven different materials-silicon, gallium arsenide, molybdenum disulfide, germanium, tin phosphate, titanium phosphate, and zirconium phosphate-under a wide variety of conditions with an RSME of 172.6 meV, which is 34% better accuracy than standard regression between the PBE and HSE band gaps. This approach, which, in principle, can be used to map PBE charge densities to band gaps or other properties computed with any higher accuracy method, has the potential to decrease computational costs, increase prediction accuracy, and enable accurate high-throughput screening for a wide variety of complex materials systems.