  A lightsail can be accelerated to ultra-high speed by the radiation pressure
of a laser having an intensity of the order of GW/m$^2$, which though presents
a critical challenge in the thermal management of lightsails. In this letter,
we explore the applicable regimes of solid-state laser cooling in dissipating
heat in additional to the previously explored radiative cooling approach. We
begin by examining the cooling capacity of laser cooling, and show that the
cooling rate from a micron-thick layer doped with ytterbium ions can exceed
that of blackbody thermal emission. This allows more intense laser illumination
upon material damage, and consequently shortened acceleration distance. Next,
we explore the impact of the limited operating bandwidth of laser cooling to
account for the Doppler shift of the pumping laser, and conclude that laser
cooling is helpful for target velocities $\lesssim5\%$ for room-temperature
operations.
