In this talk, I will report on a work in progress, which discusses gradient systems with dissipation potentials of hyperbolic-cosine type instead of the metric quadratic type. We show how such dissipation potentials emerge in large deviations of jump processes, multiscale limits of diffusion processes. We show how the exponential nature of the hyperbolic cosine derives from the exponential scaling of large deviations and arises implicitly in cell problems in multiscale limits. We discuss in-depth the role of tilting of gradient systems. We show that although in general many gradient systems are tilt-invariant, 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 structure converges to a non-quadratic gradient system of cosh-type. We show and explain how the tilt-invariance of the pre-limit system is lost in the limit system. This same lack of invariance can be recognized in classical theories of chemical reaction rates in the chemical-engineering literature. joint work with Mark Peletier (TU Eindhoven)