The NectarCAM is a camera proposed for the medium-sized telescopes in the framework of the Cherenkov Telescope Array (CTA), the next-generation observatory for very-high-energy gamma-ray astronomy. The cameras are designed to operate in an open environment and their mechanics must provide protection for all their components under the conditions defined for the CTA observatory. In order to operate in a stable environment and ensure the best physics performance, each NectarCAM will be enclosed in a slightly overpressurized, nearly air-tight, camera body, to prevent dust and water from entering. The total power dissipation will be ~7.7 kW for a 1855-pixel camera. The largest fraction is dissipated by the readout electronics in the modules. We present the design and implementation of the cooling system together with the test bench results obtained on the NectarCAM thermal demonstrator.