Measurements of $H_0$ in modified gravity theories: The role of lensed
  quasars in the late-time Universe
  In this work, we obtain measurements of the Hubble constant in the context of
modified gravity theories. We set up our theoretical framework by considering
viable cosmological $f(R)$ and $f(T)$ models, and we analyzed them through the
use of geometrical data sets obtained in a model-independent way, namely,
gravitationally lensed quasars with measured time delays, standard clocks from
cosmic chronometers, and standard candles from the Pantheon Supernovae Ia
sample. We find $H_0=(72.4\pm 1.4)$ km s$^{-1}$ Mpc$^{-1}$ and $H_0=(71.5\pm
1.3)$ km s$^{-1}$ Mpc$^{-1}$ for the $f(R)$ and $f(T)$ models, respectively.
Our results represent 1.9\% and 1.8\% measurements of the Hubble constant,
which are fully consistent with the local estimate of $H_0$ by the Hubble Space
Telescope. We do not find significant departures from general relativity, as
our study shows that the characteristic parameters of the extensions of gravity
beyond general relativity are compatible with the $\Lambda$CDM cosmology.
Moreover, within the standard cosmological framework, our full joint analysis
suggests that it is possible to measure the dark energy equation of state
parameter at 1.2\% accuracy, although we find no statistical evidence for
deviations from the cosmological constant case.
