We investigate the idea that current cosmic acceleration could be the consequence of gravitational leakage into extra dimensions on cosmological scales rather than the result of a non-zero cosmological constant, and consider the ability of future gravitational-wave siren observations to probe this phenomenon and constrain the parameters of phenomenological models of this gravitational leakage. In theories that include additional non-compact spacetime dimensions, the gravitational leakage intro extra dimensions leads to a reduction in the amplitude of observed gravitational waves and thereby a systematic discrepancy between the distance inferred to such sources from GW and EM observations. We investigate the capability of a gravitational space interferometer such as LISA to probe this modified gravity on large scales. We find that the extent to which LISA will be able to place limits on the number of spacetime dimensions and other cosmological parameters characterising modified gravity will strongly depend on the actual number and redshift distribution of sources, together with the uncertainty on the GW measurements. A relatively small number of sources ($sim 1$) and high measurement uncertainties would strongly restrict the ability of LISA to place meaningful constraints on the parameters in cosmological scenarios where gravity is only five-dimensional and modified at scales larger than about $sim 4$ times the Hubble radius. Conversely, if the number of sources observed amounts to a four-year average of $sim 27$, then in the most favourable cosmological scenarios LISA has the potential to place meaningful constraints on the cosmological parameters with a precision of $sim 1%$ on the number of dimensions and $sim 7.5%$ on the scale beyond which gravity is modified, thereby probing the late expansion of the universe up to a redshift of $sim 8$.
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