Flying ad hoc networks (FANETs) in 6G environments require routing schemes that can cope with rapid topology changes, limited UAV energy resources, and unstable air-to-air links. This paper proposes a risk-sensitive spatio-temporal graph reinforcement learning framework for predictive and propulsion-aware FANET routing. The proposed method models the FANET as a dynamic aerial graph, predicts future link states, estimates route survival probability, and selects stable routes using a risk-sensitive DRL policy. Unlike conventional GNNDRL routing, the framework jointly considers temporal topology evolution, route break risk, relay load, and communication, computation, and propulsion energy. Simulation results show that, under high mobility, the proposed method improves PDR from 0.82 to 0.90, reduces route breaks by 37.9%, extends network lifetime by 10.1%, and lowers 95th-percentile delay by 18.5% compared with the strongest baseline. These results demonstrate improved routing stability, energy efficiency, and scalability in 6G-enabled FANETs.