We report the first observation of laser cooling in Yb3+:KYW and validate the results by comparison with experiments in the well-studied material Yb3+:YAG. Radiation from a single-mode Ti:Al2O3 laser was used to achieve cooling of 1.5 K/W in 1% Yb:KYW at 1025 nm, comparing well with the reference material 3% Yb:YAG which cooled by 3.5 K/W at 1030 nm under open lab conditions. Experimental results for KYW crystals mounted on aerogels and doped with 1-20% Yb were in excellent agreement with the theoretical dependence of cooling power on the Yb absorption spectrum. Elimination of thermal conduction through the sample support structure was found to permit the attainment of lower temperatures and to simplify modeling of radiation balance conditions in self-cooled lasers with longitudinal thermal gradients. Contrary to the notion that more coolant ions yield higher cooling power, concentrations of Yb over 1% caused re-absorption of luminescence in KYW crystals, leading to a progressive red shift in the optimal cooling wavelength and the prevention of laser cooling altogether in a 20% sample at room temperature. The prospect of attaining radiation-balanced lasing in commercially-available tungstate crystals is evaluated.