We address the issue of the fair energy-efficient resource allocation for a cooperative Orthogonal Frequency Division Multiplexing (OFDM)-based cognitive radio network. In such network, the primary system leases a fraction of its subcarriers to the secondary system in exchange for the secondary users (SUs) acting as decode-and-forward relays. A class of β-fair cost function is adopted to balance the energy efficiency and fairness among SUs. In the resource allocation, we aim to optimally allocate three types of wireless resources, power, subcarriers, relay nodes, among the primary and secondary systems. Thus an optimization problem of minimizing the β-fair cost of the average transmit powers while fulfilling the rate constraints of both the primary and secondary system is formulated. This problem is solved by applying the Lagrangian dual method and a joint fair resource allocation policy at the SUs is derived in a closed-form expression. Moreover, the numerical results are provided to show the performance of the proposed resource allocation policy and the tradeoff between energy efficiency and fairness.