High-frequency measurements of the transverse susceptibility and damping constant of CoFeHfO thin films have been made over a frequency range of 0.1 GHz to 6 GHz as a function of film resistivity, thickness, and temperature. The film resistivity varied from 250 mu Ohm cm to 2100 mu Ohm cm. The films show relatively low damping at high frequencies with the damping constant a ranging from 0.01 to 0.06. The damping constant increases with film resistivity and, for the highest resistivity films, the damping constant decreases as the thickness increases. The damping constant, induced anisotropy, and film resistivity show weak temperature dependence over a temperature range from 4 K to 300 K. The low damping constant, in conjunction with the high anisotropy, large resistivity, and large spin-dependent-tunneling magnetoresistance, makes this material attractive for several high-frequency magnetic device applications.