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We study the asymptotic behavior of solutions of a stochastic timedependent damped wave equation. With the critical growth restrictions on the nonlinearity f and the time-dependent damped term, we prove the global existence of solutions and characterize their long-time behavior. We show the existence of random attractors with finite fractal dimension in H 1 0 (U ) × L 2 (U ). In particular, the periodicity of random attractors is also obtained with periodic force term and coefficient function. Furthermore, we construct the pullback random exponential attractors.