We perform fits of unconventional dark energy models to the available data from highredshift supernovae, distant galaxies and baryon oscillations. The models are based either on brane cosmologies or on Liouville strings in which a relaxation dark energy is provided by a rolling dilaton field (Q-cosmology). An interesting feature of such cosmologies is the possibility of effective four-dimensional negative-energy dust and/or exotic scaling of dark matter. An important constraint that can discriminate among models is the evolution of the Hubble parameter as a function of the redshift, H(z). We perform fits using a unifying formula for the evolution of H(z), applicable to different models. We find evidence for a negative-energy dust at the current era, as well as for exotic-scaling (a −δ ) contributions to the energy density, with 3.3 δ 4.3. The latter could be due to dark matter coupling with the dilaton in Q-cosmology models, but it is also compatible with the possibility of dark radiation from a brane Universe to the bulk in brane-world scenarios, which could also encompass Q-cosmology models. The best-fit model seems to include an a −2 -scaling contribution to the energy density of the Universe, which is characteristic of the dilaton relaxation in Q-cosmology models, not to be confused with the spatial curvature contribution of conventional cosmology. We conclude that Q-cosmology fits the data equally well with the ΛCDM model for a range of parameters that are in general expected from theoretical considerations.