We present global fits of cosmologically stable axion-like particle and QCD axion models in the mass range 0.1 neV to 10 eV. We focus on the case where the Peccei-Quinn symmetry is broken before the end of inflation, such that the initial value of the axion field can be considered to be homogeneous throughout the visible Universe. We include detailed likelihood functions from light-shining-through-wall experiments, haloscopes, helioscopes, the axion relic density, horizontal branch stars,

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... ova 1987A, white dwarf cooling, and gamma-ray observations. We carry out both frequentist and Bayesian analyses, with and without the inclusion of white dwarf cooling. We explore the degree of fine-tuning present in different models and identify parameter regions where it is possible for QCD axion models to account for both the dark matter in the Universe and the cooling hints, comparing them to specific DFSZ- and KSVZ-type models. We find the most credible parameter regions, allowing us to set (prior-dependent) upper and lower bounds on the axion mass. Our analysis also suggests that QCD axions in this scenario most probably make up a non-negligible but sub-dominant component of the dark matter in the Universe.