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For every k > 1, the kth cohomology group H^k(X, ) of the random flag complex X ∼ X(n,p) passes through two phase transitions: one where it appears, and one where it vanishes. We describe the vanishing threshold and show that it is sharp. Using the same spectral methods, we also find a sharp threshold for the fundamental group π_1(X) to have Kazhdan's property (T). Combining with earlier results, we obtain as a corollary that for every k > 3 there is a regime in which the random flag complex isarXiv:1207.0149v3 fatcat:gz2tk43b25gmvmb2kmyzwy2liu