We use the rotation-invariant Greens function method (RGM) and the high-temperature expansion (HTE) to study the thermodynamic properties of the Heisenberg antiferromagnet on the pyrochlore lattice. We discuss the excitation spectra as well as various thermodynamic quantities, such as spin correlations, uniform susceptibility, specific heat and static and dynamical structure factors. For the ground state we present RGM data for arbitrary spin quantum numbers $S$. At finite temperatures we focus on the extreme quantum cases $S=1/2$ and $S=1$. We do not find indications for magnetic long-range order for any value of $S$. We discuss the width of the pinch point in the static structure factor in dependence on temperature and spin quantum number. We compare our data with experimental results for the pyrochlore compound NaCaNi$_2$F$_7$ ($S=1$). Thus, our results for the dynamical structure factor agree well with the experimentally observed features at 3 ldots 8~meV for NaCaNi$_2$F$_7$. We analyze the static structure factor ${S}_{bf q}$ to find regions of maximal ${S}_{bf q}$. The high-temperature series of the ${S}_{bf q}$ provide a fingerprint of weak {it order by disorder} selection of a collinear spin structure, where (classically) the total spin vanishes on each tetrahedron and neighboring tetrahedra are dephased by $pi$.
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