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Cognitive radio is a promising technology to improve spectral efficiency. However, the secure performance of a secondary network achieved by using physical layer security techniques is limited by its transmit power and channel fading. In order to tackle this issue, a cognitive unmanned aerial vehicle (UAV) communication network is studied by exploiting the high flexibility of a UAV and the possibility of establishing line-of-sight links. The average secrecy rate of the secondary network is maximized by robustly optimizing the UAV's trajectory and transmit power. Our problem formulation takes into account two practical inaccurate location estimation cases, namely, the worst case and the outage-constrained case. In order to solve those challenging non-convex problems, an iterative algorithm based on $\mathcal{S}$-Procedure is proposed for the worst case while an iterative algorithm based on Bernstein-type inequalities is proposed for the outage-constrained case. The proposed algorithms can obtain effective suboptimal solutions of the corresponding problems. Our simulation results demonstrate that the algorithm under the outage-constrained case can achieve a higher average secrecy rate with a low computational complexity compared to that of the algorithm under the worst case. Moreover, the proposed schemes can improve the secure communication performance significantly compared to other benchmark schemes.