  We investigate the cosmological evolution in a new modified teleparallel
theory, called $f(T,B)$ gravity, which is formulated by connecting both $f(T)$
and $f(R)$ theories with a boundary term $B$. Here, $T$ is the torsion scalar
in teleparallel gravity and $R$ is the scalar curvature. For this purpose, we
assume flat Friedmann-Robertson-Walker (FRW) geometry filled with perfect fluid
matter contents. We study two cases in this gravity: One is for a general
function of $f(T,B)$, and the other is for a particular form of it given by the
term of $-T+F(B)$. We also formulate the general energy constraints for these
cases. Furthermore, we explore the validity of the bounds on the energy
conditions by specifying different forms of$f(T,B)$ and $F(B)$ function
obtained by the reconstruction scheme for de Sitter, power-law, the
$\Lambda$CDM and Phantom cosmological models. Moreover, the possible
constraints on the free model parameters are examined with the help of region
graphs. In addition, we explore the evolution of the effective equation of
state (EoS) $\omega_{eff}$ for the universe and compare theoretical results
with the observational data. It is found that the effective EoS represents the
phantom phase or the quintessence one in the accelerating universe in all of
the cases consistent with the observational data.
