  In the 5-dimensional braneworld cosmology, the Friedmann equation of our
4-dimensional universe on a brane is modified at high temperatures while the
standard Big Bang cosmology is reproduced at low temperatures. Based on two
well-known scenarios, the Randall-Sundrum and Gauss-Bonnet braneworld
cosmologies, we investigate the braneworld cosmological effect on the relic
density of a non-thermal dark matter particle whose interactions with the
Standard Model particles are so weak that its relic density is determined by
the freeze-in mechanism. For dark matter production processes in the early
universe, we assume a simple scenario with a light vector-boson mediator for
the dark matter particle to communicate with the Standard Model particles. We
find that the braneworld cosmological effect can dramatically alters the
resultant dark matter relic density from the one in the standard Big Bang
cosmology. As an application, we consider a right-handed neutrino dark matter
in the minimal $B-L$ extended Standard Model with a light $B-L$ gauge boson
($Z^\prime$) as a mediator. We find an impact of the braneworld cosmological
effect on the search for the long-lived $Z^\prime$ boson at the
planned/proposed Lifetime Frontier experiments.
