<p>This paper explores a novel connection between a thermodynamic and a dynamical systems perspective on emergent dynamical order. We provide evidence for a conjecture that Hamiltonian systems with mixed chaos spontaneously find regular behavior when minimally coupled to a thermal bath at sufficiently low temperature. Numerical evidence across a diverse set of five dynamical systems supports this conjecture, and allows us to quantify corollaries about the organization timescales and disruption of order at higher temperatures. Balancing the damping-induced phase-space contraction against thermal exploration, we are able to predict the transition temperatures in terms of the relaxation timescales, indicating a novel nonequilibrium fluctuation-dissipation relation, and formally connecting the thermodynamic and dynamical systems views. Our findings suggest that for a wide range of real-world systems, coupling to a cold thermal bath leads to emergence of robust, non-trivial dynamical order, rather than a mere reduction of motion as in equilibrium.</p>