From first-principles density-functional theory calculations, we show that above a given hydrostatic pressure, an unstable TO mode appears in the cubic perovskite barium stannate (BaSnO3) at Γ. Below this critical pressure, we predict an interesting lowering of the dielectric response of this compound due to a change in the sign of this TO mode effective charge. The results are compared to recent calculations on perovskites under high pressure (titanates, zirconates, and niobates), which exhibit a different mechanism at the origin of the soft mode, and to magnesium silicate (MgSiO3) that does not undergo any ferroelectric transition under hydrostatic pressure.