  We present modified cosmological scenarios that arise from the application of
the "gravity-thermodynamics" conjecture, using the Barrow entropy instead of
the usual Bekenstein-Hawking one. The former is a modification of the black
hole entropy due to quantum-gravitational effects that deform the black-hole
horizon by giving it an intricate, fractal structure. We extract modified
cosmological equations which contain new extra terms that constitute an
effective dark-energy sector, and which coincide with the usual Friedmann
equations in the case where the new Barrow exponent acquires its
Bekenstein-Hawking value. We present analytical expressions for the evolution
of the effective dark energy density parameter, and we show that the universe
undergoes through the usual matter and dark-energy epochs. Additionally, the
dark-energy equation-of-state parameter is affected by the value of the Barrow
deformation exponent and it can lie in the quintessence or phantom regime, or
experience the phantom-divide crossing. Finally, at asymptotically large times
the universe always results in the de-Sitter solution.
