  In this work, two originally separate adjustments for the Friedmann equations
are concurrently considered. Firstly, the fractal structure of the black hole
horizon region is imposed by the Barrow entropy. The second adjustment is the
$f(\mathcal{T})$ gravity, which is based on a teleparallel framework
generalization of the Einstein-Hilbert action, where $\mathcal{T}$ is the
scalar torsion. This can be considered as a Barrow entropy modification of the
$f(\mathcal{T})$ gravity thus yielding a new model. Gravity thermodynamics
hypothesis principles are used to integrate these two models under a unified
framework. We derive the modified Friedmann equation and note the corrections
obtained. To understand the implications of such dual modifications, an
application is analyzed and a particular $f(\mathcal{T})$ toy model is chosen
for the purpose. The equation of the state parameter of the resulting model,
the dimensionless density parameters of matter and dark energy, and the
deceleration parameter are explored to check the viability of the new model. A
discussion of the dynamic evolution of the universe follows from these results.
It is seen that the results comply with the observations. The newly developed
model is promising and demands further study.
