Third unramified cohomology group of a cubic threefold over a function field in one variable

We prove that the third unramified cohomology group of a smooth cubic threefold over the function field of a complex curve vanishes. For this, we combine a method of C. Voisin with Galois descent on the codimension 2 Chow group. As a corollary, we show that the integral Hodge conjecture holds for degree 4 classes on smooth projective fourfolds equipped with a fibration over a curve, the generic fibre of which is a smooth cubic threefold, with arbitrary singularities on the special fibres.