We present work in progress using the Logarithmic Linear Relaxation (LLR) density of states algorithm to analyse first-order phase transitions in pure-gauge SU($N$) Yang--Mills theories, focusing on $N = 4$ and 6. By using the LLR algorithm we aim to avoid super-critical slowing down at such transitions. Motivation for this study comes from composite dark matter models, which may feature a first-order confinement transition in the early Universe that would produce a background of gravitational waves. Improving our understanding of these phase transitions will help probe these models using observations from future gravitational-wave observatories. In addition to the confinement transition, we also analyze bulk phase transitions of the lattice theories, which feature much larger latent heat.