Role of the extra coupling in the kinetic term in Ho$ř$ava-Lifshitz gravity

We analyze different claims on the role of the coupling constant $\lambda$ in so-called $\lambda$-$R$ models, a minimal generalization of general relativity inspired by Hořava-Lifshitz gravity. The dimensionless parameter $\lambda$ appears in the kinetic term of the Einstein-Hilbert action, leading to a one-parameter family of classical theories. Performing a canonical constraint analysis for closed spatial hypersurfaces, we obtain a result analogous to that of Bellorín and Restuccia, who showed that all nonprojectable $\lambda$-$R$ models are equivalent to general relativity in the asymptotically flat case. However, the tertiary constraint present for closed boundary conditions assumes a more general form. We juxtapose this with an earlier finding by Giulini and Kiefer, who ruled out a range of $\lambda$-$R$ models by a physical, cosmological argument. We show that their analysis can be interpreted consistently within the projectable sector of Hořava-Lifshitz gravity, thus resolving the apparent contradiction.