We consider the Hamiltonian quantum electrodynamics in the Friedmann-Robertson-Walker space-time. We evaluate the path integral for the Hamiltonian QED on de Sitter background. The corresponding heat kernel expansion differs from the results of covariant approach. This difference is due to the fact that in the Hamiltonian approach to curved space QED only a part of smooth gradient transformations should be regarded as gauge ones. The Faddeev-Popov quantization scheme is modified accordingly.