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We propose a fully ab initio based integrated approach to determine the volume and temperature dependent free-energy surface of nonmagnetic crystalline solids up to the melting point. The approach is based on density-functional theory calculations with a controlled numerical accuracy of better than 1 meV/atom. It accounts for all relevant excitation mechanisms entering the free energy including electronic, quasiharmonic, anharmonic, and structural excitations such as vacancies. To achieve the desired accuracy of $<1\text{ }\text{meV}/\text{atom}$ for the anharmonic free-energy contribution without losing the ability to perform these calculations on standard present-day computer platforms, we develop a numerically highly efficient technique: we propose a hierarchical scheme---called upsampled thermodynamic integration using Langevin dynamics---which allows for a significant reduction in the number of computationally expensive ab initio configurations compared to a standard molecular dynamics scheme. As for the vacancy contribution, concentration-dependent pressure effects had to be included to achieve the desired accuracy. Applying the integrated approach gives us direct access to the free-energy surface $F(V,T)$ for aluminum and derived quantities such as the thermal expansion coefficient or the isobaric heat capacity and allows a direct comparison with experiment. A detailed analysis enables us to tackle the long-standing debate over which excitation mechanism (anharmonicity vs vacancies) is dominant close to the melting point.