The linear stabilization approach is well-known for facilitating the use of large time-step sizes while maintaining stability. However, traditional stabilization parameter selection relies on either the global Lipschitz nonlinearity or the boundedness assumption of numerical solutions. Considering the Swift--Hohenberg equation without a global Lipschitz nonlinearity, we construct a one-parameter family of third-order exponential-time-differencing Runge-Kutta (ETDRK3) scheme with the Fourier pseudo-spectral discretization, and determine the free parameter and stabilization size required to preserve energy stability. Additionally, we establish an optimal rate convergence analysis and error estimate in the norm using Sobolev embedding. The characterization of the stabilization parameter and error estimates represent significant advancements for a third-order accurate scheme applied to a gradient flow without the global Lipschitz continuity.
