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 LCDM 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 LCDM 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 and the current standard model. We introduce a diagnostic that flags such a running of H_0.