  We explore bounce cosmology in $F(\mathcal{G})$ gravity with the Gauss-Bonnet
invariant $\mathcal{G}$. We reconstruct $F(\mathcal{G})$ gravity theory to
realize the bouncing behavior in the early universe and examine the stability
conditions for its cosmological solutions. It is demonstrated that the bouncing
behavior with an exponential as well as a power-law scale factor naturally
occurs in modified Gauss-Bonnet gravity. We also derive the $F(\mathcal{G})$
gravity model to produce the ekpyrotic scenario. Furthermore, we construct the
bounce with the scale factor composed of a sum of two exponential functions and
show that not only the early-time bounce but also the late-time cosmic
acceleration can occur in the corresponding modified Gauss-Bonnet gravity.
Also, the bounce and late-time solutions in this unified model is explicitly
analyzed.
