This work focuses on the efficient numerical simulation of the Cahn-Hilliard phase-field model of diblock copolymers on evolving surfaces. The model integrates Cahn-Hilliard dynamic of diblock copolymers with partial differential equation on evolving surfaces, which has the property of geometric complexity, nonlinearity and mass conservation. In the numerical simulation, the space-time discretization of the proposed model is realized by the evolving surface finite element method. The stabilized semi-implicit approach is included in the framework of the evolving surface finite element method to produce a linear, stable, conserved and high-accurate scheme for long time numerical simulations. The stability analysis of the designed numerical method is established. Through several numerical experiments, the convergence and stability of the numerical method are investigated. In addition, spinodal decomposition is performed to research the mass evolution and dynamics of the Cahn-Hilliard model of diblock copolymers on different evolving surfaces.