Energy flow of $\lambda$ in Hořava-Lifshitz cosmology

Hořava-Lifshitz gravity has been proposed as a renormalizable and ghost-free quantum gravity model candidate with an anisotropic UV-scaling between space and time. We present here a cosmological background analysis of two different formulations of the theory, with particular focus on the running of the parameter $\lambda$. Using a large dataset consisting of cosmic microwave background data from Planck, $\text{Pantheon}+\text{supernovae}$ catalog, SH0ES Cepheid variable stars, baryon acoustic oscillations (BAO), cosmic chronometers, and gamma-ray bursts (GRB), we arrive at new bounds on the cosmological parameters, in particular $\lambda$, which runs logarithmically with energy and describes deviation from general relativity, which corresponds to $\lambda =1$ For the detailed balance scenario we arrive at the bound $\lambda =1.0406\pm 0.0023$, and for beyond detailed balance the limit reads $\lambda =1.0064\pm 0.0002$. We also study the influence of different datasets and priors, and we find that removing low-redshift data generally moves $\lambda$ closer toward UV values, while simultaneously widening the error bars. In the detailed balance scenario, this effect is more noticeable, and without prior $\lambda \ge 1$ it takes on values that are significantly below unity.