Gauss-Bonnet inflation with a constant rate of roll

We consider the constant-roll condition in the model of the inflaton nonminimal coupling to the Gauss-Bonnet term. By assuming the first Gauss-Bonnet flow parameter $\delta_1$ is a constant, we discuss the constant-roll inflation with constant $\epsilon_1$, constant $\epsilon_2$ and constant $\eta_H$, respectively. Using the Bessel function approximation, we get the analytical expressions for the scalar and tensor power spectrum and derive the scalar spectral index $n_{\mathcal{R}}$ and the tensor to scalar ratio $r$ to the first order of $\epsilon_1$. By using the Planck 2018 observations constraint on $n_{\mathcal{R}}$ and $r$, we obtain some feasible parameter space and show the result on the $n_{\mathcal{R}}-r$ region. The scalar potential is also reconstructed in some spectral cases.