We present a new accelerated distributed algorithm for the robust solution of convex optimization problems over networks. We propose a novel distributed restarting mechanism for accelerated optimization dynamics with individual asynchronous time-varying coefficients. Graph-dependent restarting conditions are derived to establish suitable stability, convergence, and robustness properties for problems characterized by strongly convex smooth and non-smooth primal functions. Since the algorithm combines continuous-time dynamics and discrete-time dynamics, we model the complete system as a hybrid dynamical system. Numerical results illustrate our results.