We consider the standard first passage percolation model in ℤd for d≥2. We are interested in two quantities, the maximal flow τ between the lower half and the upper half of the box, and the maximal flow ϕ between the top and the bottom of the box. A standard subadditive argument yields the law of large numbers for τ in rational directions. Kesten and Zhang have proved the law of large numbers for τ and ϕ when the sides of the box are parallel to the coordinate hyperplanes: the two variables grow linearly with the surface s of the basis of the box, with the same deterministic speed. We study the probabilities that the rescaled variables τ/s and ϕ/s are abnormally small. For τ, the box can have any orientation, whereas for ϕ, we require either that the box is sufficiently flat, or that its sides are parallel to the coordinate hyperplanes. We show that these probabilities decay exponentially fast with s, when s grows to infinity. Moreover, we prove an associated large deviation principle of speed s for τ/s and ϕ/s, and we improve the conditions required to obtain the law of large numbers for these variables.