  In this paper, using Barrow entropy, we propose an interacting model of
Barrow holographic dark energy (BHDE). In particular, we study the evolution of
a spatially flat FLRW universe composed of pressureless dark matter and BHDE
that interact with each other through a well-motivated interaction term.
Considering the Hubble horizon as the IR cut-off, we then study the
evolutionary history of important cosmological parameters, particularly, the
density parameter, the equation of state parameter, and the deceleration
parameter in the BHDE model and find satisfactory behaviors in the model. We
perform a detailed study on the dynamics of the field equations by studying the
asymptotic behavior of the field equations, while we write the analytic
expression for the scale factor with the use of Laurent series. Finally, we
study the implications of gravitational thermodynamics in the interacting BHDE
model with the dynamical apparent horizon as the cosmological boundary. In
particular, we study the viability of the generalized second law by assuming
that the apparent horizon is endowed with Hawking temperature and Barrow
entropy.
