This paper proposes a distributed gradient dynamics approach for solving Multi-Objective AC Optimal Power Flow (MO-ACOPF) problem. The fuel cost, environmental pollution, and network loss are considered as objective functions. The non-convex MO-ACOPF problem is reformulated as a gradient dynamical system. We derive condition under which the equilibrium point of the proposed gradient dynamical system is the saddle point of the MO-ACOPF Lagrangian function. We propose a distributed solution algorithm for derived MO-ACOPF gradient dynamical system. In the distributed solution algorithm, each bus in the power network is assumed as a local agent, which only uses information from its neighbor agents to compute optimal operating points. We prove convergence of our distributed algorithm to the equilibrium point of the MO-ACOPF gradient dynamical system (and accordingly to the primal and dual optimum of original MO-ACOPF). Accordingly, our proposed approach has the following main advantages: (a) it is a distributed and decentralized algorithm and accordingly it needs less computational time as compared to the centralized model, (b) it has the proof of convergence to the global optimal solution, and (c) it is suitable for varying loads in power networks. The simulation results based on standard IEEE 14-, 30-, 39-, 57-, 118- and 300-bus test systems in different scenarios are carefully studied. The proposed distributed gradient dynamics provides the best results compared to the relevant literature as illustrated by simulation results.