In this paper, we study power allocation in cognitive sensor networks where cognitive users (cognitive enabled sensor nodes) opportunistically share a common spectrum with primary users (licensed devices). We define an energy efficiency-oriented utility function as a new metric to evaluate power allocation. Consider that sensor nodes are self-interested to maximize their own utility, we formulate the energy efficient power allocation problem as a non-cooperative game. We firstly prove that there exist Nash equilibriums in the proposed game. Secondly, we prove that the power allocation game is a supermodular game with some conditions. Finally, we use best response algorithm to identify the Nash equilibrium. Simulations are conducted to demonstrate that the proposed power allocation strategy can achieve satisfactory performance in terms of energy efficiency, convergence speed and fairness in cognitive sensor networks.