We perform fits of unconventional dark energy models to the available data from high-redshift supernovae and differential galaxy ages. 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 δ≳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.