We report on the low-temperature magnetoelastic properties of the skyrmion-lattice compound GdRu₂Si₂ under magnetic fields applied along the [001] axis. Our sound-velocity measurements reveal a series of pronounced anomalies related to the skyrmion-lattice phase. The emergence of hysteresis and steplike features indicates intrinsic metastability of the skyrmion lattice in GdRu₂Si₂, deviating from other skyrmion-lattice compounds. Analysis of multiple elastic modes allows us to disentangle the compressional and shear responses, with the latter showing a stronger sensitivity to the metastability of the skyrmion lattice.