A new model of the energy selective electron (ESE) device composed of four electron reservoirs and four energy filters is proposed. The device can work as a thermal amplifier, which may be equivalent to a coupling system consisting of two three-electron-reservoir heat pumps. Expressions for the heat-pumping rate (HPR) and coefficient of performance (COP) of the device are analytically derived through the heat flows of two-terminal ESE devices. The maximum HPR and COP are calculated. It is found that the maximum HPR of the coupling system can attain 2 times of that of a three-electron-reservoir heat pump while its COP is still equal to that of a three-electron-reservoir heat pump. Moreover, the effects of the chemical potentials of electron reservoirs, central levels of energy filters, half width at half maximum of energy filters on the HPR and COP of the device are discussed in detail. The optimal configuration of the device is determined. The cut-off value of the half width at half maximum of energy filters is calculated. The optimum selection criteria of main parameters are provided.