Hořava-Lifshitz gravity has been proposed as a ghost-free and renormalizable quantum gravity model candidate with an anisotropic UV-scaling between space and time. In my talk I would like to present a cosmological background analysis of two different 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. In the two mentioned cosmological models the obtained bounds differ from the value λ=1 corresponding to Einstein gravity and seem to run with the energies of the data sources.