Remember the cryoscopic constant (given in mole fraction) is
K' = (RT^2)/H of fusion....where R is the gas constant, T is the standard freezing point of pure substance and H of fusion is the enthalpy of fusion...the only thing that is variable is the enthalpy of fusion. However, in many cases, you can assume that enthalpy is constant by the use of Trouton's rule that says that the standard entropy of vaporization is approximately 85J/K mol.
So no...the cryoscopic constant does not vary with temperature. Enthalpy does vary with temperature but you can assume it to be constant.
