  Recently there has been a surge of interest in studying Lorentzian quantum
cosmology using Picard-Lefschetz methods. The present paper aims to explore the
Lorentzian path-integral of Gauss-Bonnet gravity in four spacetime dimensions
with metric as the field variable. We employ mini-superspace approximation and
study the variational problem exploring different boundary conditions. It is
seen that for mixed boundary conditions non-trivial effects arise from
Gauss-Bonnet sector of gravity leading to additional saddle points for lapse in
some case. As an application of this we consider the No-boundary proposal of
the Universe with two different settings of boundary conditions, and compute
the transition amplitude using Picard-Lefschetz formalism. In first case the
transition amplitude is a superposition of a Lorentzian and a Euclidean
geometrical configuration leading to interference incorporating
non-perturbative effects coming from Gauss-Bonnet sector of gravity. In the
second case involving complex initial momentum we note that the transition
amplitude is an analogue of Hartle-Hawking wave-function with non-perturbative
correction coming from Gauss-Bonnet sector of gravity.
