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In conventional WLANs, stations (STAs) select access points (APs) using the existing scheme based on the current quality of links. However, some novel improved architecture of WLANs is proposed with cognitive radio (CR) to negotiate spectrum usage, where the existing AP selection scheme might not be suitable for it. Thus in this paper, a new optimal AP Selection based on Restless Bandits (APSRB) scheme with the "indexability" property is proposed for WLANs with CR to maximize the throughput and to minimize the energy consumption. The AP selection problem is firstly established as a restless bandit problem, which is solved by the primal-dual index heuristic algorithm based on the first order relaxation with low complexity to yield APSRB scheme. Additionally, the APSRB scheme is divided into offline computation and online selection, where main work will be finished in former one so as to decrease the complexity further. Finally, extensive simulation results illustrate the significant performance improvement of the APSRB scheme compared to the existing one in different scenarios.