Unmanned aerial vehicles (UAVs) can provide wireless access services to terrestrial users without geographical limitations and will become an essential part of the future communication system. However, the openness of wireless channels and the mobility of UAVs make the security of UAV-based communication systems particularly challenging. This work investigates the security of aerial cognitive radio networks (CRNs) with multiple uncertainties colluding eavesdroppers. A cognitive aerial base station transmits messages to cognitive terrestrial users using the spectrum resource of the primary users. All secondary terrestrial users and illegitimate receivers jointly decode the received message. The average secrecy rate of the aerial CRNs is maximized by jointly optimizing the UAV's trajectory and transmission power. An iterative algorithm based on block coordinate descent and successive convex approximation is proposed to solve the non-convex mixed-variable optimization problem. Numerical results verify the effectiveness of our proposed algorithm and show that our scheme improves the secrecy performance of airborne CRNs.