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This article establishes cutoff stability also known as abrupt thermalizationfor generic multidimensional Hurwitz stable Ornstein-Uhlenbeck systems with(possibly degenerate) L\'evy noise at fixed noise intensity. The results arebased on several ergodicity quantitative lower and upper bounds some of whichmake use of the recently established shift linearity property of theWasserstein-Kantorovich-Rubinstein distance by the authors. It covers suchirregular systems like Jacobi chains and more general networks of coupledharmonic oscillators with a heat bath (including L\'evy excitations) atconstant temperature on the outer edges and the so-called Brownian gyrator.

Ergodicity bounds for stable Ornstein-Uhlenbeck systems in Wasserstein distance with applications to cutoff stability