We present the power spectrum of the reconstructed halo density field derived from a sample of luminous red galaxies (LRGs) from the Sloan Digital Sky Survey (SDSS) Seventh Data Release (DR7). The halo power spectrum has a direct connection to the underlying dark matter power for k≤ 0.2 h Mpc−1, well into the quasi-linear regime. This enables us to use a factor of ∼8 more modes in the cosmological analysis than an analysis with kmax= 0.1 h Mpc−1, as was adopted in the SDSS team analysis of the DR4 LRG sample. The observed halo power spectrum for 0.02 < k < 0.2 h Mpc−1 is well fitted by our model: χ2= 39.6 for 40 degrees of freedom for the best-fitting Λ cold dark matter (ΛCDM) model. We find Ωmh2(ns/0.96)1.2= 0.141+0.010−0.012 for a power-law primordial power spectrum with spectral index ns and Ωbh2= 0.022 65 fixed, consistent with cosmic microwave background measurements. The halo power spectrum also constrains the ratio of the comoving sound horizon at the baryon-drag epoch to an effective distance to z= 0.35: rs/DV(0.35) = 0.1097+0.0039−0.0042. Combining the halo power spectrum measurement with the Wilkinson Microwave Anisotropy Probe (WMAP) 5 year results, for the flat ΛCDM model we find Ωm= 0.289 ± 0.019 and H0= 69.4 ± 1.6 km s−1 Mpc−1. Allowing for massive neutrinos in ΛCDM, we find eV at the 95 per cent confidence level. If we instead consider the effective number of relativistic species Neff as a free parameter, we find Neff= 4.8+1.8−1.7. Combining also with the Kowalski et al. supernova sample, we find Ωtot= 1.011 ± 0.009 and w=−0.99 ± 0.11 for an open cosmology with constant dark energy equation of state w. The power spectrum and a module to calculate the likelihoods are publicly available at http://lambda.gsfc.nasa.gov/toolbox/lrgdr/.