In this paper, we investigate the power allocation of primary base station (PBS) and cognitive base station (CBS) across different orthogonal frequency division multiplexing (OFDM) subcarriers for energy-efficient secure downlink communication in OFDM-based cognitive radio networks (CRNs) with the existence of an eavesdropper having multiple antennas. We propose a secrecy energy efficiency maximization (SEEM) scheme by exploiting the instantaneous channel state information (ICSI) of the eavesdropper, called ICSI based SEEM (ICSI-SEEM) scheme with a given total transmit power budget for different OFDM subcarriers of both PBS and CBS. As for the case when the eavesdropper's ICSI is unknown, we also propose an SEEM scheme through using the statistical CSI (SCSI) of the eavesdropper, namely SCSI based SEEM (SCSI-SEEM) scheme. Since the ICSI-SEEM and SCSI-SEEM problems are fractional and non-convex, we first transform them into equivalent subtractive problems, and then achieve approximate convex problems through employing the difference of two-convex functions approximation method. Finally, new two-tier power allocation algorithms are proposed to achieve $\varepsilon$-optimal solutions of our formulated ICSI-SEEM and SCSI-SEEM problems. Simulation results illustrate that the ICSI-SEEM has a better secrecy energy efficiency (SEE) performance than SCSI-SEEM, and moreover, the proposed ICSI-SEEM and SCSI-SEEM schemes outperform conventional SR maximization and EE maximization approaches in terms of their SEE performance.