We investigate two new observational perspectives in the context of torsional
gravitational modification of general relativity, i.e., the $f(T)$ gravity: i)
We use Pantheon data of type Ia supernovae motivated by a time variation of the
Newton's constant on the supernovae distance modulus relation, and find that a
joint analysis with Baryon Acoustic Oscillations (BAO) and Big Bang
Nucleosynthesis (BBN), i.e., Pantheon+BAO+BBN, provides constraints on the
effective free parameter of the theory to be well compatible with the
$\mathrm{\Lambda}$ CDM prediction; ii) We present the framework of $f(T)$ gravity at the
level of linear perturbations with the phenomenological functions, namely the
effective gravitational coupling $\mu $ and the light deflection parameter
$\mathrm{\Sigma}$ , which are commonly used to parameterize possible modifications of the
Poisson equation relating the matter density contrast to the lensing and the
Newtonian potentials, respectively. We use the available Cosmic Microwave
Background (CMB) data sets from the Planck 2018 release to constrain the free
parameters of the $f(T)$ gravity and $\mathrm{\Lambda}$ CDM models. We find that CMB
data, and its joint analyses with Pantheon and BAO data constrain the $f(T)$
gravity scenario to be practically indistinguishable from the $\mathrm{\Lambda}$ CDM
model. We obtain the strongest limits ever reported on $f(T)$ gravity scenario
at the cosmological level.