  We reverse the logic behind the apparent existence of $H_0$-tension, to
design diagnostics for cosmological models. The basic idea is that the
non-constancy of $H_0$ inferred from observations at different redshifts is a
null hypothesis test for models within the FLRW paradigm -- if $H_0$ runs, the
model is wrong. Depending on the kind of observational data, the most suitable
form of the diagnostic can vary. As examples, we present two $H_0$ diagnostics
that are adapted to two different BAO observables. We use these and the
corresponding BAO data to Gaussian reconstruct the running of $H_0$ in flat
$\Lambda$CDM with Planck values for the model parameters. For flat $\Lambda$CDM
when the radiation contribution can be neglected, with comoving distance data,
the diagnostic is a simple hypergeometric function. Possible late time
deviations from the FLRW paradigm can also be accommodated, by simply keeping
track of the (potentially anisotropic) sky variation of the diagnostic.
