  We construct Barrow holographic dark energy in the case of non-flat universe.
In particular, considering closed and open spatial geometry we extract the
differential equations that determine the evolution of the dark-energy density
parameter, and we provide the analytical expression for the corresponding dark
energy equation-of-state parameter. We show that the scenario can describe the
thermal history of the universe, with the sequence of matter and dark energy
epochs. Comparing to the flat case, where the phantom regime is obtained for
relative large Barrow exponents, the incorporation of positive curvature leads
the universe into the phantom regime for significantly smaller values.
Additionally, in the case of negative curvature we find a reversed behavior,
namely for increased Barrow exponent we acquire algebraically higher
dark-energy equation-of-state parameters. Hence, the incorporation of slightly
non-flat spatial geometry to Barrow holographic dark energy improves the
phenomenology while keeping the new Barrow exponent to smaller values.
