Running Hubble Tension and a H0 Diagnostic
  Hubble tension is routinely presented as a mismatch between the Hubble
constant $H_0$ determined locally and a value inferred from the flat
$\Lambda$CDM cosmology. In essence, the tension boils down to a disagreement
between two numbers. Here, assuming the tension is cosmological in origin, we
predict that within flat $\Lambda$CDM there should be other inferred values of
$H_0$, and that a "running of $H_0$ with redshift" can be expected. These
additional determinations of $H_0$ may be traced to a difference between the
effective equation of state (EoS) of the Universe within the
Friedmann-Lema\^itre-Robertson-Walker (FLRW) cosmology framework and the
current standard model. We introduce a diagnostic that flags such a running of
$H_0$.
