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In this article, a high-order time-stepping scheme based on the cubic interpolation formula is considered to approximate the generalized Caputo fractional derivative (GCFD). Convergence order for this scheme is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>4</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mspace width="3.33333pt"></mspace><mo>(</mo><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> is the order of the GCFD. The local truncation error is also provided. Then, we adopt the developed scheme to establish a difference scheme for the solution of the generalized fractional advection–diffusion equation with Dirichlet boundary conditions. Furthermore, we discuss the stability and convergence of the difference scheme. Numerical examples are presented to examine the theoretical claims. The convergence order of the difference scheme is analyzed numerically, which is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>4</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> in time and second-order in space.