Ensuring the desired security is an important issue in all communicating systems and it becomes more challenging in open medium networks. Furthermore, the majority of the wireless and mobile networks including Flying Adhoc NETworks (FANETs) are built over energy-restricted devices. Hence, communication interface and computation complexity are additional factors to consider when designing secure protocols. as an alternative solution to cryptography, trust and reputation management solutions are able to sustain high security levels with less energy consumption. In this paper, we propose a new reputation-aware energy-efficient FANET monitoring technique, in which we first establish trust among different peers and then thanks to the trust and links duration combination, we distribute the network monitoring period between those trusting each other moving with similar mobility patterns. Simulations conducted using NS2 simulator show the efficiency of our proposal at both ensuring high detection rations (> 80%) and reducing by more than two times the average consumed energy.