Flying Ad-Hoc Network (FANET) consists of unmanned aerial vehicles (UAVs) that cooperatively establish wireless communication networks. It has been widely used in communications, transportation, forest fire monitoring and other fields. Due to the highly dynamic topology and boundary disconnection problems in three-dimensional (3-D) environments, designing an efficient routing protocol for FANETs remains challenging. To address these issues, this paper proposes a Boundary-Aware Routing Protocol utilizing Game Theory (BARP-GT). Firstly, BARP-GT defines a 3-D forwarding area by considering network boundary effects. Then, it estimates node degree in different boundary scenarios without relying on Hello message broadcasting, which reduces routing overhead and energy consumption. In addition, the forwarding process of route request packets is modeled as a static game to derive an adaptive forwarding probability and suppress broadcast storms. Simulation results based on Network Simulator-2 (NS-2) demonstrate that BARP-GT achieves better performance than iPipe, QEHLR, and BLPR in terms of end-to-end delay, packet delivery ratio, packet drop ratio, and routing load.