We use a combination of three computational methods to investigate the notoriously difficult frustrated three-dimensional pyrochlore quantum antiferromagnet, at finite temperature : canonical typicality for a finite cluster of unit cells (i.e., 32 sites), a finite- matrix product state method on a larger cluster with 48 sites, and the numerical linked cluster expansion (NLCE) using clusters up to 25 lattice sites, including nontrivial hexagonal and octagonal loops. We calculate thermodynamic properties (energy, specific heat capacity, entropy, susceptibility, magnetization) and the static structure factor. We find a pronounced maximum in the specific heat at , which is stable across finite size clusters and converged in the series expansion. At (the limit of convergence of our method), the residual entropy per spin is , which is relatively large compared to other frustrated models at this temperature. We also observe a nonmonotonic dependence on of the magnetization at low magnetic fields, reflecting the dominantly nonmagnetic character of the low-energy states. A detailed comparison of our results to measurements for the material yields a rough agreement of the functional form of the specific heat maximum, which in turn differs from the sharper maximum of the heat capacity of the spin ice material .