<p>We perform a state-of-the-art study of the cosmological phase transitions of the real-scalar extended Standard Model. We carry out a broad scan of the parameter space of this model at next-to-next-to-leading order in powers of couplings. We use effective field theory to account for the necessary higher-order resummations, and to construct consistent real and gauge-invariant gravitational wave predictions. Our results provide a comprehensive account of the convergence of perturbative predictions for the gravitational wave signals in this model. For the majority of the parameter points in our study, we observe apparent convergence. While leading and next-to-leading order predictions of the gravitational wave amplitude typically suffer from relative errors between 10 and 10⁴, at next-to-next-to-leading order the typical relative errors are reduced to between 0<i>.</i>5 and 50. Nevertheless, for those parameter points predicting the largest signals, potentially observable by future gravitational wave observatories, the validity of the perturbative expansion is in doubt.</p>