Running Hubble tension and a H0 diagnostic

Hubble tension is routinely presented as a mismatch between the Hubble constant H₀ determined locally and a value inferred from the flat 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 there should be other inferred values of H₀, and that a ``running of H₀ with redshift'' can be expected. These additional determinations of H₀ may be traced to a difference between the effective equation of state (EoS) of the Universe within the Friedmann-Lema\^tre-Robertson-Walker (FLRW) cosmology framework and the current standard model. We introduce a diagnostic that flags such a running of H₀.