Due to the rapid increase in the usage and demand of wireless sensor networks (WSN), the limited frequency spectrum available for WSN applications will be extremely crowded in the near future. More sensor devices also mean more recharging/replacement of batteries, which will cause significant impact on the global carbon footprint. In this paper, we propose a relay-assisted cognitive radio sensor network (CRSN) that allocates communication resources in an environmentally friendly manner. We use shared band amplify and forward relaying for cooperative communication in the proposed CRSN. We present a multi-objective optimization architecture for resource allocation in a green cooperative cognitive radio sensor network (GC-CRSN). The proposed multi-objective framework jointly performs relay assignment and power allocation in GC-CRSN, while optimizing two conflicting objectives. The first objective is to maximize the total throughput, and the second objective is to minimize the total transmission power of CRSN. The proposed relay assignment and power allocation problem is a non-convex mixed-integer non-linear optimization problem (NC-MINLP), which is generally non-deterministic polynomial-time (NP)-hard. We introduce a hybrid heuristic algorithm for this problem. The hybrid heuristic includes an estimation-of-distribution algorithm (EDA) for performing power allocation and iterative greedy schemes for constraint satisfaction and relay assignment. We analyze the throughput and power consumption tradeoff in GC-CRSN. A detailed analysis of the performance of the proposed algorithm is presented with the simulation results.