**9. Conclusions**

We analyzed the PS together with the BAOs in three bins in both the ΛCDM and *w*0*wa*CDM models to investigate if an evolutionary trend of *H*<sup>0</sup> persists also with the contribution of BAOs and by varying two parameters contemporaneously with *H*<sup>0</sup> (Ω0*<sup>m</sup>* and *w<sup>a</sup>* for the ΛCDM and *w*0*wa*CDM, respectively). The persistence of the trend of *H*<sup>0</sup> as a function of redshift is also shown in the case of the Hu–Sawicki model. We here stress that the main goal of the current analysis is to highlight the reliability of the trend of *H*0(*z*) and not to further constrain Ω0*<sup>m</sup>* or any other cosmological parameters. With the subsequent fitting of *H*<sup>0</sup> values through the model *g*(*z*) = H0/(1 + *z*) *η* , we obtain

*<sup>η</sup>* <sup>∼</sup> <sup>10</sup>−<sup>2</sup> , as in the previous work [51]: those are compatible with zero from 1.2 to 5.8 *σ* (see Table 1). The multidimensional results could reveal a dependence on the redshift of *H*0, assuming that it is observable at any redshift scale. If this evolution is not caused by statistical effects and other selection biases or hidden evolution of SNe Ia parameters [364], we show how *H*0(*z*) could modify the luminosity distance definition within the modified theory of gravity. If we consider a theoretical interpretation for the observed trend, new cosmological scenarios may explain an evolving Hubble constant with the redshift. For instance, we test in Sections 6.2 and 7 a simple class of modified gravity theories given by the *f*(*R*) models in the equivalent scalar-tensor formalism. In principle, this could be due to an effective varying Einstein constant governed by a slow evolution of a scalar field which mediates the gravity-matter interaction. However, the slow decreasing trend of *H*<sup>0</sup> has proven to be independent of the Hu–Sawicki model application. Indeed, if this theory had worked we would have observed the trend of the *η* parameter to be flattened out and be compatible with 0 in 1 *σ* at any redshift bin. This is not the case, thus new scenarios must be explored within the modified theories of gravity or slightly alternative approaches (see Section 8.2). We can state that this evolving trend of *H*<sup>0</sup> is independent of the starting values of the fitting for *H*<sup>0</sup> (we here have considered *H*<sup>0</sup> = 70) and, thus, on the fiducial *M* and on the redshift bins and even when we consider two cosmological parameters changing contemporaneously (Ω0*<sup>m</sup>* and *w<sup>a</sup>* in ΛCDM and *w*0*wa*CDM models, respectively). Thus, we need to further investigate the nature of this trend. In addition, the implementation of GRBs as cosmological probes together with SNe Ia and BAOs has proven to be not only possible in a near future but also necessary since the redshift range that GRBs cover is much larger than the one typical of SNe Ia. This last characteristic will surely allow GRBs to give further information on the nature of the early universe and pose new constraints in the future measurements of *H*0.

**Author Contributions:** M.G.D. performed the conceptualization of all project, data curation, formal analysis, methodology, writing original draft, validation, supervision, software. B.D.S. performed data curation, visualization, formal analysis, methodology, writing original draft, software. T.S. performed formal analysis, visualization, methodology, writing original draft on f(R) and revised it, G.M. performed a partial conceptualization limited to the theoretical part of the f(R) gravity theory; E.R. edited and review the analysis on f(R) and participated in the general discussion and conceptualization of the paper. G.L. wrote Section 6.2.1 and gave suggestions on the cosmological constraints on *w*. M.B. performed the conceptualization on the priors to answer the referee report. S.U. performed a formal analysis on changing the parameters together with *H*0. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This work made use of Pantheon sample data [284], which can be found in the GitHub repository: https://github.com/dscolnic/Pantheon (accessed on 21 December 2020). This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. We are thankful to V. Nielson, A. Lenart, G. Sarracino, and D. Jyoti for their support on cosmological computations. T. Schiavone is supported in part by INFN under the program TAsP (Theoretical Astroparticle Physics). G.L. and B.D.S. acknowledge the support of INFN. T.S. acknowledges the support of the Department of Physics of the University of PISA. M.G.D. acknowledges the support from NAOJ and NAOJ—Division of Science.

**Conflicts of Interest:** The authors declare no conflict of interest.
