Thermal noise is ubiquitous in microscopic systems and in high-precision measurements. Controlling thermal noise, especially using laser light to apply dissipation, substantially affects science in revealing the quantum regime of gases 1 , in searching for fundamental physics 2 , and in realizing practical applications 3 . Recently, nonlinear light-matter interactions in microresonators have opened up new classes of microscopic devices. A key example is Kerrmicroresonator frequency combs 4 ; so-called soliton microcombs not only explore nonlinear science but also enable integrated-photonics devices, such as optical synthesizers 5 , optical clocks 6 , and data-communications systems 7 . Here, we explore how thermal noise leads to fundamental decoherence of soliton microcombs. We show that a particle-like soliton exists in a state of thermal equilibrium with its silicon-chip-based resonator. Therefore the soliton microcomb's modal linewidth is thermally broadened. Our experiments utilize record sensitivity in carrier-envelope-offset frequency detection in order to uncover this regime of strong thermal-noise correlations. Furthermore, we have discovered that passive laser cooling of the soliton reduces thermal decoherence to far below the ambient-temperature limit. We implement laser cooling by microresonator photothermal forcing, and we observe cooling of the frequency-comb light to an effective temperature of 84 K. Our work illuminates inherent connections between nonlinear photonics, microscopic physical fluctuations, and precision metrology that could be harnessed for innovative devices and methods to manipulate light.