A novel energy-stable scheme is proposed to solve the spatial fractional Cahn–Hilliard equations, using the idea of scalar auxiliary variable (SAV) approach and stabilization technique. Thanks to the stabilization technique, it is shown that larger temporal stepsizes can be applied in numerical simulations. Moreover, the proposed SAV finite difference scheme is non-coupled and linearly implicit, which can be efficiently solved by the preconditioned conjugate gradient (PCG) method with a sine transform based preconditioner. Optimal error estimates of the fully-discrete scheme are obtained rigorously. Numerical examples are given to confirm the theoretical results and show the higher efficiency of the proposed scheme than the previous SAV schemes.