Fundamental thermodynamic considerations reveal that efficient emission from an electrically injected light emitting diode (LED) can lead to the cooling of the device. This effect, known as electroluminescent (EL) cooling, has been identified decades ago, but it has not been experimentally demonstrated in semiconductors at practical operating conditions due to the extreme requirements set for the efficiency of the light emission. To probe the conditions of EL cooling in GaAs based light emitters, we have designed and fabricated LED structures with integrated photodiodes (PDs), where the optically mediated thermal energy transport between the LED and the PD can be easily monitored. This allows characterization of the fundamental properties of the LED and a path for eliminating selected issues encountered in conventional approaches for EL cooling, such as the challenging light extraction. Despite several remaining nonidealities, our setup demonstrates a very high directly measured quantum efficiency of 70%. To characterize the bulk part of the LED, we also employ a model for estimating the power conversion efficiency (PCE) of the LED, without the contribution of non-fundamental nonidealities such as photodetection losses. Our results suggest that the PCE of the LED peaks at around 105-115%, exceeding the 100% barrier required to reach the EL cooling regime by a clear margin. This implies that the LED component in our device is in fact cooling down by transporting thermal energy carried by the emitted photons to the PD. This provides a compelling incentive for further study to confirm the result and to find ways to extend it for practically useful EL cooling.