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 ΛCDM 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 ΛCDM 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-Lemaître-Robertson-Walker (FLRW) cosmology framework and the current standard model. We introduce a diagnostic that flags such a running of H0.