We discuss the cosmological degeneracy between the Hubble parameter H(z), the age of the universe and cosmological parameters describing simple variations from the minimal LCDM model. We show that independent determinations of the Hubble parameter H(z) such as those recently provided by Simon et al., Phys. Rev. D 71 (2005) 123001, combined with Cosmic Microwave Background data, can provide stringent constraints on possible deviations from the LCDM model. In particular we find that this data combination constrains at the 68% (95%) confidence level the following parameters: sum of the neutrino masses < 0.5 (1.0) eV, number of relativistic neutrino species N_rel = 4.1^{+0.4}_{-0.9} (^{+1.1}_{-1.5}), dark energy equation of state parameter w = -0.95 +- 0.17 (+- 0.32), and curvature Ω_k = 0.002 +- 0.006 (+- 0.014), in excellent agreement with dataset combinations involving Cosmic Microwave Background, Supernovae and Baryon Acoustic Oscillations. This offers a valuable consistency check for systematic errors. Comment: 10 pages, 7 figures. Added references. Minor changes added to match accepted version in JCAP