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We review a class of gradient systems with dissipation potentials of hyperbolic-cosine type. We show how such dissipation potentials emerge in large deviations of jump processes, multi-scale limits of diffusion processes, and more. We show how the exponential nature of the cosh derives from the exponential scaling of large deviations and arises implicitly in cell problems in multi-scale limits. We discuss in-depth the role of "tilting" of gradient systems. Certain classes of gradient systems are "tilt-independent", which means that changing the driving functional does not lead to changes of the dissipation potential. Such tilt-independence separates the driving functional from the dissipation potential, guarantees a clear modelling interpretation, and gives rise to strong notions of gradient-system convergence. We show that although in general many gradient systems are tilt-independent, certain cosh-type systems are not. We also show that this is inevitable, by studying in detail the classical example of the Kramers high-activation-energy limit, in which a diffusion converges to a jump process and the Wasserstein gradient system converges to a cosh-type system. We show and explain how the tilt-independence of the pre-limit system is lost in the limit system. This same lack of independence can be recognized in classical theories of chemical reaction rates in the chemical-engineering literature. We illustrate a similar lack of tilt-independence in a discrete setting. For a class of "two-terminal" fast subnetworks, we give a complete characterization of the dependence on the tilting, which strongly resembles the classical theory of equivalent electrical networks.