Significance Climate models are foundational to formulations of climate policy and must successfully reproduce key features of the climate system. The temporal spectrum of observed global surface temperature is one such critical benchmark. This spectrum is known to obey scaling laws connecting astronomical forcings, from orbital to annual scales. We provide evidence that the current hierarchy of climate models is capable of reproducing the increase in variance in global-mean temperature at low frequencies. We suggest that successful climate predictions at decadal-to-centennial horizons hinge critically on the accuracy of initial and boundary conditions, particularly for the deep ocean state.