Hořava-Lifshitz gravity is  perturbatively renormalizable quantum gravity model candidate with an anisotropic UV-scaling between space and time. I would like to present a cosmological background analysis of the minimal projectable formulations of the theory, with particular focus on the running of the parameter λ with energy. Using a large dataset consisting of Cosmic Microwave Background data from Planck, Pantheon+ supernovae catalogue, SH0ES Cepheid variable stars, Baryon acoustic oscillations (BAO), Cosmic Chronometers, and gamma-ray bursts (GRB), provide new bounds on the cosmological parameters, in particular λ, which describes deviation from the classical general relativity. The obtained bounds differ from the value λ=1 corresponding to Einstein gravity and seem to run with the energies of the data sources.