Here we present an investigation into how cooling of the plasma influences the oscillation properties (e.g., eigenfunctions and eigenfrequencies) of transverse (i.e., kink) magnetohydrodynamic (MHD) waves in a compressible magnetic flux tube embedded in a gravitationally stratified and uniformly magnetized atmosphere. The cooling is introduced via a temperature-dependent density profile. A time-dependent governing equation is derived and an approximate zeroth-order solution is then obtained. From this the influence of cooling on the behavior of the eigenfrequencies and eigenfunctions of the transverse MHD waves is determined for representative cooling timescales. It is shown analytically, as the loop cools, how the amplitude of the perturbations is found to decrease as time increases. For cooling timescales of 900-2000 s (as observed in typical EUV loops), it is shown that the cooling has important and relevant influence on the damping times of loop oscillations. Next, the theory is put to the test. The damping due to cooling is fitted to a representative observation of standing kink oscillation of EUV loops. It is also shown with an explicit approximate analytical form, how the period of the fundamental and first harmonic of the kink mode changes with time as the loop cools. A consequence of this is that the value of the period ratio P 1/P 2, a tool that is popular in magneto-seismological studies in coronal diagnostics, decreases from the value of a uniform loop, 2, as the temperature decreases. The rate of change in P 1/P 2 is dependent upon the cooling timescale and is well within the observable range for typical EUV loops. Further to this, the magnitude of the anti-node shift of the eigenfunctions of the first harmonic is shown to continually increase as the loop cools, giving additional impetus to the use of spatial magneto-seismology of the solar atmosphere. Finally, we suggest that measurements of the rate of change in the eigenfunctions and eigenfrequencies of MHD oscillations can provide values for the cooling timescale and a further insight into the physics of coronal loops.