In this study, we consider the Cahn--Hilliard (CH) equation with the arbitrary polynomial
formula degenerate mobility, which is a function valued mobility depending on the concentration. Because the CH equation with the degenerate mobility relates to a various scientific topics, such as logarithm potential and the curvature-dependent mobility, it is important to confirm the dynamics and construct a numerical scheme, which satisfies a unique solvability, unconditional energy stability, and mass preservation. Accordingly, we propose an unconditionally energy gradient stable scheme based upon the linear stabilized splitting scheme. the energy dissipation and mass preservation properties of the CH equation with an arbitrary polynomial formula degenerate mobility are proved at first. Subsequently, the discrete mass preservation property, unique solvability, energy gradient stability, and accuracy of the considered numerical scheme are numerically analyzed and convicted from the numerical experiments.