This paper studies the energy efficiency (EE) maximization for an orthogonal frequency division multiple access (OFDMA) downlink network aided by a relay station (RS) with subcarrier pairing. A highly flexible transmission protocol is considered, where each transmission is executed in two time slots. Every subcarrier in each slot can either be used in direct mode or be paired with a subcarrier in another slot to operate in relay mode. The resource allocation (RA) in such a network is highly complicated, because it has to determine the operation mode of subcarriers, the assignment of subcarriers to users, and the power allocation of the base station and RS. We first propose a mathematical description of the RA strategy. Then, a RA algorithm is derived to find the globally optimum RA to maximize the EE. Finally, we present extensive numerical results to show the impact of minimum required rate of the network, the user number, and the relay position on the maximum EE of the network.