We investigate electron transport in asymmetric double-barrier (Al, Ga)As/GaAs thermionic cooling heterostructures. Measurements of temperature-dependent current-voltage characteristics confirm that the dominant electron transport is a sequential process of resonant tunneling injection into and thermionic emission from the quantum-well (QW) cooling layer. The thermal activation energy of the current is found to be strongly dependent on the bias voltage. Furthermore, instead of showing a simple thermal activation behavior, the current exhibits rather complicated temperature and voltage dependence, particularly when the thermionic emission barrier is low. To establish a quantitative understanding, we develop an intuitive analytical model for sequential electron transport that explicitly takes into account scattering effects in the thermionic emission process from the two-dimensional QW states to the three-dimensional above-barrier states. The observed temperature-dependent sequential current is well explained by the present theory.