We show that there is a sharp threshold in dimension one for the transport cost between the Lebesgue measure $λ$ and an invariant random measure $μ$ of unit intensity to be finite. We show that for \emph{any} such random measure the $L^1$ cost are infinite provided that the first central moments $\mathbb{E}[|n-μ([0,n))|]$ diverge. Furthermore, we establish simple and sharp criteria, based on the variance of $μ([0,n)]$, for the $L^p$ cost to be finite for $0<p<1$.