In this article, we aim to design a task scheduling scheme for the underwater multiobjective task of target hunting and environmental search. A distributed autonomous underwater vehicle (AUV) network is deployed to perform the task, where AUVs equipped with sensors can cooperatively search the environment and hunt the target by sharing local information. To achieve efficient exploration of the overall environment by the AUV network, we design an intranetwork cooperative searching approach based on the Age of Information (AoI). Besides, it is critical to conceive an energy-efficient mechanism due to the energy constraints of AUVs and the difficulty of sustainable energy supply. To address the aforementioned issues, we propose an energy-efficient distributed multiagent proximal policy optimization (DMAPPO) scheme to perform real-time AUV target hunting and environment searching in underwater turbulent fields. The proposed scheme can adjust the number of AUVs assigned to each objective according to practical requirement and residual energy. Distributed AUVs can make decisions autonomously and cooperatively complete the task efficiently through limited information interaction. In addition, we derive a lower bound on the policy improvement of MAPPO. Moreover, our simulation results demonstrate that the proposed scheme outperforms the standard algorithms in terms of hunting efficiency, degree of searching, and network energy efficiency.