  We explore cosmology with a bounce in Gauss-Bonnet gravity where the
Gauss-Bonnet invariant couples to a dynamical scalar field. In particular, the
potential and and Gauss-Bonnet coupling function of the scalar field are
reconstructed so that the cosmological bounce can be realized in the case that
the scale factor has hyperbolic and exponential forms. Furthermore, we examine
the relation between the bounce in the string (Jordan) and Einstein frames by
using the conformal transformation between these conformal frames. It is shown
that in general, the property of the bounce point in the string frame changes
after the frame is moved to the Einstein frame. Moreover, it is found that at
the point in the Einstein frame corresponding to the point of the cosmological
bounce in the string frame, the second derivative of the scale factor has an
extreme value. In addition, it is demonstrated that at the time of the
cosmological bounce in the Einstein frame, there is the Gauss-Bonnet coupling
function of the scalar field, although it does not exist in the string frame.
