The spectrum sensing performance in cooperative cognitive radio (CR) network is studied under a double threshold (DTH)-based detection with censoring of CRs, and thereafter, the study is extended for a hybrid spectrum access scheme in presence of Rayleigh faded sensing (S) and reporting (R) channels. In spectrum sensing, a CR employs an energy detection to detect the presence of primary user (PU) and compares the received energy statistics with the DTH. The CRs with energy statistics lying in fuzzy zone are not allowed to send their sensing information to the fusion centre (FC). Further, the qualified CRs are censored (rank-based and threshold-based censoring) to report their decisions based on quality of R-channel. The incorporation of DTH-based sensing and censoring of CRs not only improves the detection performance but also reduces the transmission overhead. In spectrum access, two hybrid spectrum access schemes, namely conventional hybrid spectrum access scheme (CHSAS) and a modified hybrid spectrum access scheme (MHSAS) are studied and compared.