Unmanned aerial vehicles (UAVs) are becoming a popular research topic in applications that do not require human intervention. A variety of applications and devices coexist in the environment where UAVs operate, resulting in a serious spectrum shortage. Therefore, cognitive radio (CR) is a promising solution for opportunistic access to underutilized spectrum bands by the primary user (PU) through cooperative spectrum sensing (CSS) technique. However, the flexible location of UAVs makes CSS inefficient and even difficult to be implemented. In view of this, a cognitive UAV network model consisting of a pair of UAVs which follows a circular flight trajectory to participate in CSS is proposed in a spectrum sensing frame structure. According to the local energy detection, we further propose an optimization problem about the stopping time in a quickest detection paradigm and conduct out Bayesian detection method with feedback to minimize the sensing delay and the false alarm probability by optimizing the stopping time. Moreover, we theoretically derive the optimal threshold pair and prove the optimal stopping time by means of Markov process. At last, a series of numerical simulations are shown to corroborate the proposed Bayesian detection method with feedback, in terms of the false alarm probability, the sensing delay, and achievable throughput. In contrast to the classic Neyman‐Pearson and Bayesian detection methods, the advantage of Bayesian detection method with feedback sensing is presented.