  We construct a generalized interacting model of Barrow Holographic Dark
Energy (BHDE) with infrared cutoff being given by the Hubble horizon. We
analyze the cosmological evolution of a flat
Friedmann-Lema\^itre-Robertson-Walker Universe filled by pressureless dark
matter, BHDE and radiation fluid. The interaction between the dark sectors of
the cosmos is assumed of non-gravitational origin and satisfying the second law
of thermodynamics and Le Chatelier-Braun principle. We study the behavior of
various model parameters, such as the BHDE density parameter, the equation of
state parameter, the deceleration parameter, the jerk parameter and the square
of sound speed. We show that our model satisfactorily retraces the thermal
history of the Universe and is consistent with current observations for certain
values of parameters, providing an eligible candidate to describe dark energy.
We finally explore the thermodynamics of our framework, with special focus on
the validity of the generalized second law.
