Bayesian Compressive Sensing (BCS), introduced into wideband cognitive radio network (CRN), has been considered as a promising technique for its ability of accurately recovering a signal from far fewer samples than required by the Nyquist sampling theorem. However, as BCS algorithm modulates the number of measurements step by step through evaluating the error bars, it needs appreciable amounts of the intermediate computation to meet the accuracy requirements of the recovery signal. In this paper, adaptive step size selection based Bayesian compressive spectrum sensing (ABCS) is proposed to reduce the recovery times for alleviating the hardware requirements of the CR receiver. In this scheme, we study the relation model between the error bar and current measurements which may not accurately recover the original signals. And then we utilize this model to select step size adaptively in order to decide the dimension of the compressive matrix, which is the increment of the used measurements. Compared to classical BCS algorithm, the simulation results show the ABCS algorithm can decrease the time of recovery signals tremendously.