We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on $\mathbb{R}^N$ describing $\nabla\varphi$ interfaces with convex potentials. Under minimal assumptions on the potential, we find that the spectral gap of the process is always given by $\mathrm{gap}_N=1-\cos(π/N)$, and that for all $ε\in(0,1)$, its $ε$-mixing time satisfies $T_N(ε)\sim \frac{\log N}{2\mathrm{gap}_N}$ as $N\to\infty$, thus establishing the cutoff phenomenon. The results reveal a universal behavior in that they do not depend on the choice of the potential.