The role of Lorentz invariance as a fundamental symmetry of nature has been lately reconsidered in different approaches to quantum gravity. It is thus natural to study whether other puzzles of physics may be solved within these proposals. This may be the case for the cosmological constant problem. Indeed, it has been shown that breaking Lorentz invariance provides Lagrangians that can drive the current acceleration of the universe without experiencing large corrections from ultraviolet physics. In this work, we focus on the simplest model of this type, called Theta CDM, and study its cosmological implications in detail. At the background level, this model cannot be distinguished from Lambda CDM. The differences appear at the level of perturbations. We show that in Theta CDM, the spectrum of CMB anisotropies and matter fluctuations may be affected by a rescaling of the gravitational constant in the Poisson equation, by the presence of extra contributions to the anisotropic stress, and finally by the existence of extra clustering degrees of freedom. To explore these modifications accurately, we modify the Boltzmann code class. We then use the parameter inference code MONTE PYTHON to confront Theta CDM with data from WMAP-7, SPT and WiggleZ. We obtain strong bounds on the parameters accounting for deviations from Lambda CDM. In particular, we find that the discrepancy between the gravitational constants appearing in the Poisson and Friedmann equations is constrained at the level of 1.8%.