We consider k‐dimensional random simplicial complexes generated from the binomial random (k + 1)‐uniform hypergraph by taking the downward‐closure. For 1 ≤ j ≤ k − 1, we determine when all cohomology groups with coefficients in urn:x-wiley:rsa:media:rsa20857:rsa20857-math-0001 from dimension one up to j vanish and the zero‐th cohomology group is isomorphic to urn:x-wiley:rsa:media:rsa20857:rsa20857-math-0002. This property is not deterministically monotone for this model, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. We also study the asymptotic distribution of the dimension of the j‐th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced by Linial and Meshulam, previously only known for dimension two.