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Ho\v{r}ava-Lifshitz gravity has been proposed as a ghost-free quantum gravitymodel candidate with an anisotropic UV-scaling between space and time. Wepresent here a cosmological background analysis of two different formulationsof the theory, with particular focus on the running of the parameter $\lambda$.Using a large dataset consisting of Cosmic Microwave Background data from {\itPlanck}, Pantheon+ supernovae catalogue, SH0ES Cepheid variable stars, Baryonacoustic oscillations (BAO), Cosmic Chronometers, and gamma-ray bursts (GRB),we arrive at new bounds on the cosmological parameters, in particular$\lambda$, which describes deviation from classical general relativity. For thedetailed balance scenario we arrive at the bound $\lambda=1.02726\pm0.00012$,and for beyond detailed balance the limit reads$\lambda=0.9949^{+0.0045}_{-0.0046}$. We also study the influence of differentdata sets and priors, and we find that removing low-redshift data generallymoves $\lambda$ closer towards UV values, whilst simultaneously widening theerror bars. In the detailed balance scenario, this effect is more noticeable,and $\lambda$ takes on values that are significantly below unity, whichcorresponds to the infrared limit of the theory.

On the energy flow of $\lambda$ in Ho\v{r}ava-Lifshitz cosmology