  Teleparallel gravity offers a new avenue in which to construct gravitational
models beyond general relativity. While teleparallel gravity can be framed in a
way to be dynamically equivalent to general relativity, its modifications are
mostly not equivalent to the traditional route to modified gravity. $f(T,B)$
gravity is one such gravitational theory where the second and fourth order
contributions to the field equations are decoupled. In this work, we explore
the all important cosmological perturbations of this new framework of gravity.
We derive the gravitational propagation equation, its vector perturbation
stability conditions, and its scalar perturbations. Together with the matter
perturbations, we derive the effective gravitational constant in this
framework, and find an interesting branching behaviour that depends on the
particular gravitational models being probed. We close with a discussion on the
relation of these results with other gravitational theories.
