  The slow-roll inflation which took place at extremely high energy regimes is
in general believed to be sensitive to the high-order curvature corrections to
the classical general relativity (GR). In this paper, we study the effects of
the high-order curvature term, the Gauss-Bonnet (GB) term, on the primordial
scalar and tensor spectra of the slow-roll inflation in the consistent $D \to
4$ Einstein Gauss-Bonnet (4EGB) gravity. The GB term is incorporated into
gravitational dynamics via the re-scaling of the GB coupling constant $\alpha
\to \alpha/(D-4)$ in the limit $ D\to 4$. For our purpose, we calculate
explicitly the primordial scalar and tensor power spectra with GB corrections
accurate to the next-to-leading order in the slow-roll approximation in the
slow-roll inflation by using the third-order uniform asymptotic approximation
method. The corresponding spectral indices and their runnings of the spectral
indices for both the scalar and tensor perturbations as well as the ratio
between the scalar and tensor spectra are also calculated up to the
next-to-leading order in the slow-roll expansions. These results represent the
most accurate results obtained so far in the literature. In addition, by
studying the theoretical predictions of the scalar spectral index and the
tensor-to-scalar ratio with Planck 2018 constraint in a model with power-law
potential, we show that the second-order corrections are important in future
measurements.
