 Solution

The cosmological principle gives spatial homogeneity and cosmological isotropy for the large-scale spatial slices. About any chosen comoving observer, spherical coordinates therefore separate a radial direction from the angular directions. In particular,is the metric tensor on the unit sphere. Its appearance expresses cosmological isotropy, rather than a privileged centre of the universe. The factor $(1-K{r}^2{)}^{-1}$ permits the radial geometry to have nonzero spatial curvature of an FLRW universe.

The scale factor $a(t)$ changes every spatial separation between fixed comoving coordinates by the same multiplicative factor. For example, the radial proper distance in cosmology between fixed radial labels at a fixed cosmic time isExpansion or contraction through this single function preserves spatial homogeneity and cosmological isotropy; direction-dependent scale factors would generally spoil the latter. The Friedmann-Lemaître-Robertson-Walker metric specifies this possible evolution geometrically; a dynamical theory is needed to determine $a(t)$.

The constant $K$ fixes the sign and normalization of the spatial curvature of an FLRW universe. The spatial scalar curvature is ${}^{(3)}\mathcal{R}=6K/a^2(t)$. With dimensionless $r$ and a suitably normalized scale factor, the familiar choices are $K=+1,0,-1$ for spherical, flat, and hyperbolic spatial geometry. A rescaling of $r$ changes the normalization of $K$ and $a$ together; its sign has invariant geometric significance. The local metric tensor alone does not fix global topology.

The coordinates share a cosmological, comoving construction: $r,\theta ,\varphi$ label the worldlines of the smooth cosmic fluid, and $t$ is the common cosmic time measured as proper time along those worldlines. Observers following that fluid keep their spatial labels while the scale factor changes their physical separations. The coordinate $t$ itself increases; it is not another constant spatial label. Thus the question's final phrase is understood as asking for their common adaptation to the cosmological fluid, rather than suggesting that all four coordinates are time independent or have the same units.