In this paper, we propose a novel censor-based cooperative spectrum sensing strategy, called adaptive energy-efficient sensing (AES), in which both sequential sensing and censoring report mechanism are employed, aiming to reduce the sensing energy consumption of secondary user relays (SRs). In AES, an anchor secondary user (SU) requires cooperative sensing only when it does not detect the presence of PU by itself, and the cooperative SR adopts decision censoring report only if the sensing result differs from its previous one. We derive the generalized-form expressions false alarm and detection probabilities over Rayleigh fading channels for AES. The sensing energy consumption is also analyzed. Then, we study sensing energy overhead minimization problem and show that the sensing time allocation can be optimized to minimize the miss detection probability and sensing energy overhead. Finally, numerical results show that the proposed strategy can remarkably reduce the sensing energy consumption while only slightly degrading the detection performance compared with traditional scheme.