  In this paper, we delve into the gravitational path integral of Gauss-Bonnet
gravity in four spacetime dimensions, in the mini-superspace approximation. Our
primary focus lies in investigating the transition amplitude between distinct
boundary configurations. Of particular interest is the case of Robin boundary
conditions, known to lead to a stable Universe in Einstein-Hilbert gravity,
alongside Neumann boundary conditions. To ensure a consistent variational
problem, we supplement the bulk action with suitable surface terms. This study
leads us to compute the necessary surface terms required for Gauss-Bonnet
gravity with the Robin boundary condition, which wasn't known earlier.
Thereafter, we perform an exact computation of the transition amplitude.
Through $\hbar\to0$ analysis, we discover that the Gauss-Bonnet gravity
inherently favors the initial configuration, aligning with the Hartle-Hawking
no-boundary proposal. Remarkably, as the Universe expands, it undergoes a
transition from the Euclidean (imaginary time) to the Lorentzian signature
(real time). To further reinforce our findings, we employ a saddle point
analysis utilizing the Picard-Lefschetz methods. The saddle point analysis
allows us to find the initial configurations which lead to Hartle-Hawking
no-boundary Universe that agrees with the exact computations. Our study
concludes that for positive Gauss-Bonnet coupling, initial configurations
corresponding to the Hartle-Hawking no-boundary Universe gives dominant
contribution in the gravitational path-integral.
