I present first results from ongoing lattice investigations into the finite-temperature dynamics of stealth dark matter, which adds to the standard model a new SU(4) gauge sector with four moderately heavy fundamental fermions. This work by the Lattice Strong Dynamics Collaboration builds on past studies of direct detection and collider searches for stealth dark matter, by analyzing the early-universe SU(4) confinement transition, which produces a stochastic background of gravitational waves if it is first order. In addition to delineating the parameter space in which a first-order transition is observed, I discuss the quantities we are analyzing in order to predict the resulting gravitational-wave spectrum.