In this paper, fundamental limits on the energy efficiency of cognitive radio transmissions are studied in the presence of statistical quality of service (QoS) constraints when the data arrival process at the cognitive transmitter is modeled as a two-state Markov chain. It is assumed that cognitive secondary users initially sense the channel via energy detection and adjust their transmission power levels according to the sensing decisions. Considering the true states of the primary user activity, imperfect sensing decisions, and the reliability of the transmissions, a state-transition model with four states is constructed for the cognitive radio channel. A framework for determining the maximum average arrival rates that can be supported in the cognitive radio channel under QoS constraints is provided by employing the notions of effective bandwidth of Markov arrivals and effective capacity of cognitive radio transmissions. After formulating the maximum average arrival rates, minimum energy per bit and wideband slope expressions are obtained in order to identify the energy efficiency of cognitive radio systems.