  In standard cosmology, the cosmic homogeneity scale is the transition scale
above which the patterns arising from non-uniformities -- such as groups and
clusters of galaxies, voids, and filaments -- become indistinguishable from a
random distribution of sources. Recently, different groups have investigated
the feasibility of using such a scale as a cosmological test and arrived at
different conclusions. In this paper, we complement and extend these studies by
exploring the evolution of the spatial (${\cal{R}}_H$) and angular ($\theta_H$)
homogeneity scales with redshift, assuming a spatially flat, $\Lambda$-Cold
Dark Matter %($\Lambda$CDM) universe and linear cosmological perturbation
theory. We confirm previous results concerning the non-monotonicity of
${\cal{R}}_H$ with the matter density parameter $\Omega_{m0}$ but also show
that it exhibits a monotonical behavior with the Hubble constant $H_0$ within a
large redshift interval. More importantly, we find that, for $z \gtrsim 0.6$,
the angular homogeneity scale not only presents a monotonical behavior with
$\Omega_{m0}$ and $H_0$ but is quite sensitive to $H_0$, especially at higher
redshifts. These results, therefore, raise the possibility of using $\theta_H$
as a new, model-independent way to constrain cosmological parameters.
