The emergent spectra of close-in, giant exoplanets ("hot Jupiters") are believed to be distinct from those of young gas giants and brown dwarfs with similar effective temperatures because these objects are primarily heated from above by their host stars rather than internally from the release of energy from their formation (Showman et al. 2020). Theoretical models predict a continuum of dayside spectra for hot Jupiters as a function of irradiation level, with the coolest planets having absorption features in their spectra, intermediate-temperature planets having emission features due to thermal inversions, and the hottest planets having blackbody-like spectra due to molecular dissociation and continuum opacity from the H- ion (Fortney et al. 2008, Parmentier et al. 2018, Arcangeli et al. 2018). Absorption and emission features have been detected in the spectra of a number of individual hot Jupiters (Kreidberg et al. 2014, Mikal-Evans et al. 2020), and population-level trends have been observed in photometric measurements (Keating et al. 2019, Baxter et al. 2020, Garhart et al. 2020, Dransfield et al. 2020). However, there has been no unified, population-level study of the thermal emission spectra of hot Jupiters such as has been done for brown dwarfs (Manjavacas et al. 2019) and transmission spectra of hot Jupiters (Sing et al. 2016). Here we show that hot Jupiter secondary eclipse spectra centered around a water absorption band at 1.4 microns follow a common trend in water feature strength with temperature. The observed trend is broadly consistent with the predictions of self-consistent one-dimensional models for how the thermal structures of solar composition planets vary with irradiation level. Nevertheless, the ensemble of planets exhibits significant scatter around the mean trend. The spread can be accounted for if the planets have modest variations in metallicity and/or elemental abundance ratios, which is expected from planet formation models (Mordasini et al. 2016, Ali-Dib et al. 2017, Madhusudhan et al. 2017, Cridland et al. 2019).