  We propose a new model of $D=4$ Gauss-Bonnet gravity. To avoid the usual
property of the integral over the standard $D=4$ Gauss-Bonnet scalar becoming a
total derivative term, we employ the formalism of metric-independent
non-Riemannian spacetime volume elements which makes the D=4 Gauss-Bonnet
action term non-trivial without the need to couple it to matter fields unlike
the case of ordinary D=4 Gauss-Bonnet gravity models. The non-Riemannian volume
element dynamically triggers the Gauss-Bonnet scalar to be an arbitrary
integration constant M on-shell, which in turn has several interesting
cosmological implications: (i) It yields specific solutions for the Hubble
parameter and the Friedmann scale factor as functions of time, which are
completely independent of the matter dynamics, i.e., there is no back reaction
by matter on the cosmological metric; (ii) For M>0 it predicts a
"coasting"-like evolution immediately after the Big Bang, and it yields a late
universe with dynamically produced dark energy density given through M; (iii)
For the special value M=0 we obtain exactly a linear "coasting" cosmology; (iv)
For M<0 we have in addition to the Big Bang also a Big Crunch with
"coasting"-like evolution around both; (v) It allows for an explicit analytic
solution of the pertinent Friedmann and scalar field equations of motion, while
dynamically fixing uniquely the functional dependence of the scalar potential.
