Physical-layer security from an energy-efficient perspective is of crucial importance in cognitive radio (CR). A CR network is considered where a secondary user (SU) coexists with a primary user in the presence of an eavesdropper and channel fading. Secure energy efficiency (EE) maximisation problems are formulated in secure green CR based on the condition that a minimum secrecy capacity (SC) of a SU is guaranteed. A peak interference power constraint and an average (ATP)/peak transmit power (PTP) constraint are imposed in the SU's Tx. Using fractional programming and the Lagrange dual method, energy-efficient optimal power allocation strategies are proposed to efficiently solve the secure EE maximisation problems. It is shown that the secure EE of the SU achieved under the ATP constraint is higher than that obtained under the PTP constraint. The tradeoff is elucidated between the secure EE and the SC of the SU.