  The possibility that the $H_0$ tension is a sign of a physics beyond the
$\Lambda$CDM model is one of the most exciting possibilities in modern
cosmology. The challenge of solving this problem is complicated by several
factors, including the worsening of the tension on $\sigma_8$ parameter when
that on $H_0$ is raised. Furthermore, the perspective from which the problem is
viewed should not be underestimated, since the tension on $H_0$ can also be
interpreted as a tension on the size of the acoustic horizon, $r_s$, which
deserves proper discussion. The common approach in the literature consists in
proposing a new model that can resolve the tension and treat the new parameters
of the theory as free in the analysis. However, allowing additional parameters
to vary often results in larger uncertainties on the inferred cosmological
parameters, causing an apparent relaxing in the tension due to the broaden in
the posterior, instead of a genuine shift in the central value of $H_{0}$. To
avoid this, we consider here an empirical approach that assumes specific
non-standard values of the $\Lambda$CDM extensions and we analyze how the
important parameters in the context of the tension vary accordingly. For our
purposes, we study simple extensions of the standard cosmological model, such
as a phantom DE component (with Equation of State $w < -1$) and extra
relativistic species in the early Universe (so that $N_{eff} > 3.046$). We
obtain relations between variation in the value of $w$ and $N_{eff}$ and
changes in $H_0$, $r_s$ and $\sigma_8$. In this way an empirical relation
between $H_0$ and $\sigma_8$ is provided, that is a first step in understanding
which classes of theoretical models, and with which characteristics, could be
able to break the correlation between the two tensions.
