Cognitive radio is a promising technology to improve spectral efficiency. However, communication security of a secondary network is limited by its transmit power and channel fading. In order to tackle this issue, by exploiting the high flexibility and the possibility of establishing line-of-sight links, a cognitive unmanned aerial vehicle (UAV) communication network is studied. The average secrecy rate of the secondary network is maximized by robustly optimizing the UAVs trajectory and transmit power. Our formulated problem takes into account practical imperfect location estimation. To solve the non-convex problem, an iterative suboptimal algorithm based on the Bernstein-type inequalities is presented. Our simulation results demonstrate that the proposed scheme can improve the secure communication performance significantly compared to a benchmark scheme based on fixed trajectory.