Although the fast kink oscillation, as one of a few fundamental modes in coronal seismology, has received a lot of attention over the past two decades, observations of its frequency drift remain elusive. There is evidence that this phenomenon is related to the quasi-static evolution of loop density. We therefore consider analytically the effects of a quasi-static density evolution on the fast kink oscillation of coronal loops. From the analyses, we determine explicitly the analytic dependence of the oscillation period/frequency and amplitude on the evolving density of the oscillatory loop. The findings can well reconcile several key characters in some frequency drift observations, which are not understood. Models of fast kink oscillation in the thermal dynamic loop are also established to investigate the present effects in more detail. Our findings not only show us a possible explanation for the frequency drift of the coronal loop's fast kink oscillation, but also a full new energy transformation mechanism where the internal energy and the kinetic energy of an oscillating coronal loop can be interchanged directly by the interaction of the loop's oscillation and its density evolution, which we suggest may provide a new clue for the energy processes associated with a thermodynamic resonator in the space magnetic plasma.