The root cause of different oscillatory behavior of turbulent stress anisotropy under rapid rotation of initially axisymmetric turbulence is theoretically investigated. For this, based on the velocity spectral tensor of axisymmetric turbulence, the rapid part of the pressure-strain is determined and the equation of the turbulent stress anisotropy is solved for initial conditions generated by axisymmetric expansion and contraction of isotropic turbulence. As is well known, the damping of turbulent stress anisotropy under rapid rotation is observed for both initial conditions, and this feature is attributed to the linear rapid rotation effect on turbulence. On the other hand, the oscillatory development of turbulent stress anisotropy can be seen conspicuously only for the initial turbulence generated by axisymmetric expansion. This selective oscillatory feature is found to be strongly related to the total strain that is applied to the isotropic turbulence to generate the initial axisymmetric turbulence. And, through an asymptotic approach, it is also found that the material frame-indifference principle of two-dimensional turbulence is the underlying physics in this different oscillatory behavior.