<p>This paper introduces a reduced-order Legendre–Galerkin (LG) extrapolation (ROLGE) method combined with the scalar auxiliary variable (SAV) approach (ROLGE-SAV) to numerical solve the Cahn–Hilliard (CH) equation. To begin with, the CH equation is reformulated by using the SAV approach as a linearized version in temporal direction with its energy non-increasing property proven. Next, LG method is considered and error estimates for the fully discrete scheme are derived, along with proofs of discrete energy non-increasing and boundedness of temporal difference quotients. To improve computational efficiency, a reduced-order model (ROM) with exptrapolation is proposed by approximating the discrete space via a low-dimensional space spanned by the proper orthogonal decomposition (POD) basis. This POD basis stems from a small set of snapshots of fully discrete solutions over an initial short interval <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\([0,T_{0}] (T_{0}\ll T)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>≪</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The exptrapolation technique is then applied to evolve the solutions over the interval <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\([T_{0},T]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. The ROLGE-SAV method reduces computational load efficiently by eliminating redundancy while preserving original solution space properties. Key advantages include unconditional stability, error estimates, and ease of implementation, all rigorously verified via numerical examples.</p>