  We study the Barrow cosmological model, which proposes that quantum gravity
effects create a complex, fractal structure for the universe's apparent
horizon. We leverage the thermodynamics - gravity conjecture. By applying the
Clausius relation to the apparent horizon of the Friedmann - Lema\^itre -
Robertson - Walker universe within this framework, we derive modified field
equations where the Barrow entropy is linked to the horizon. We assess the
Barrow cosmology against current observations - cosmic microwave background ,
supernovae , and baryon acoustic oscillations data - and include projections
for future Laser Interferometer Space Antenna (LISA) standard sirens (SS). Our
numerical results suggest a modest improvement in the Hubble tension for Barrow
cosmology with phantom dark energy behavior, compared to the standard
cosmological model. Furthermore, incorporating simulated LISA SS data alongside
existing observational constraints tightens the limitations on cosmological
parameters, particularly the deformation exponent.
