Energy consumption is a major challenge in IoT devices, which was aimed to be improved by employing energy-efficient objective functions (OFs) in the structure of the RPL routing protocol. Meanwhile, the majority of the existing OFs mainly perform the parent selection based on the gathered information from the routing layer. Nevertheless, based on our investigations, there exists a series of transmission operations in the medium access control (MAC) layer, which significantly affects the energy consumption in IoT devices. Therefore, in this article, we propose ELITE, an energy-efficient cross-layer OF, which introduces a novel routing metric, called strobe per packet ratio (SPR). SPR indicates the number of transmitted strobes per packet due to radio duty cycling (RDC) policies in the MAC layer. This newly defined metric, which has been designed to be coupled with asynchronous MAC protocols, could be differentiated node by node and based on the existing relative phase shift between the communicating nodes. In this regard, the ELITE tries to select a path, which imposes less number of strobe transmissions to its nodes. According to the evaluation results, while ELITE could reduce the average amount of required strobes per packet by up to 25%, it can significantly improve the average amount of consumed energy in an IoT node by up to 39% compared to its counterpart OFs.