  We investigate the evolution of gravitational waves through discontinuous
evolution (transition) of the Hubble expansion rate $H(z)$ at a sudden
cosmological singularity, which may be due to a transition of the value of the
gravitational constant. We find the evolution of the scale factor and the
gravitational wave waveform through the singularity by imposing the proper
boundary conditions. We also use existing cosmological data and mock data of
future gravitational wave experiments (the ET) to impose current and
anticipated constraints on the magnitude of such a transition. We show that
mock data of the Einstein Telescope can reduce the uncertainties by up to a
factor of three depending on the cosmological parameter considered.
