  Recently, it was formulated a teleparallel theory called $f(T,B)$ gravity
which connects both $f(T)$ and $f(R)$ under suitable limits. In this theory,
the function in the action is assumed to depend on the torsion scalar $T$ and
also on a boundary term related with the divergence of torsion,
$B=2\nabla_{\mu}T^{\mu}$. In this work, we study different features of a flat
Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) cosmology in this theory. First,
we show that the FLRW equations can be transformed to the form of Clausius
relation $\hat{T}_hS_{\rm eff}=-dE+WdV$, where $\hat{T}_h$ is the horizon
temperature and $S_{\rm eff}$ is the entropy which contains contributions both
from horizon entropy and an additional entropy term introduced due to the
non-equilibrium. We also formulate the constraint for the validity of the
generalised second law of thermodynamics (GSLT). Additionally, using a
cosmological reconstruction technique, we show that both $f(T,B)$ and $-T+F(B)$
gravity can mimic power-law, de-Sitter and $\Lambda$CDM models. Finally, we
formulate the perturbed evolution equations and analyse the stability of some
important cosmological solutions.
