We study a system of N particles interacting through the Kac collision, with m of them interacting, in addition, with a Maxwellian thermostat at temperature 1β. We use two indicators to understand the approach to the equilibrium Gaussian state. We prove that (i) the spectral gap of the evolution operator behaves as mN for large N and (ii) the relative entropy approaches its equilibrium value (at least) at an eventually exponential rate ∼mN2 for large N. The question of having non-zero entropy production at time 0 remains open. A relationship between the Maxwellian thermostat and the thermostat used in the work of Bonetto et al. [J. Stat. Phys. 156(4), 647–667 (2014)] is established through a Van Hove limit.