  The gradient flow provides a new class of renormalized observables which can
be measured with high precision in lattice simulations. In principle this
allows for many interesting applications to renormalization and improvement
problems. In practice, however, such applications are made difficult by the
rather large cutoff effects found in many gradient flow observables. At lowest
order of perturbation theory we here study the leading cutoff effects in a
finite volume gradient flow coupling with SF and SF-open boundary conditions.
We confirm that O($a^2$) Symanzik improvement is achieved at tree-level,
provided the action, observable and the flow are O($a^2$) improved. O($a^2$)
effects from the time boundaries are found to be absent at this order, both
with SF and SF-open boundary conditions. For the calculation we have used a
convenient representation of the free gauge field propagator at finite flow
times which follows from a recently proposed set-up by L\"uscher and renders
lattice perturbation theory more practical at finite flow time and with SF,
open, SF-open or open-SF boundary conditions.
