  The $S=1/2$ hyperkagome-lattice Heisenberg antiferromagnet, which for
instance is related to the experimentally accessible spinel oxide
Na$_4$Ir$_3$O$_8$, allows to study the interplay of geometrical frustration and
quantum as well as thermal fluctuations in three dimensions. We use 16 terms of
a high-temperature series expansion complemented by the entropy-method
interpolation to examine the specific heat and the uniform susceptibility of
the $S=1/2$ hyperkagome-lattice Heisenberg antiferromagnet. We obtain
thermodynamic quantities for the two possible scenarios of either a gapless or
a gapped energy spectrum. We have found that the specific heat $c$ exhibits,
besides the high-temperature peak around $T\approx 0.669$, a low-temperature
one at $T\approx 0.021\ldots0.033$. The functional form of the uniform
susceptibility $\chi$ below about $T=0.5$ depends strongly on whether the
energy spectrum is gapless or gapped. The value of the ground-state energy can
be estimated to $e_{0}\approx-0.440\ldots-0.435$. In addition to the
entropy-method interpolation we use the finite-temperature Lanczos method to
calculate $c$ and $\chi$ for finite lattices of $N=24$ and $36$ sites. A
combined view on both methods leads us to favour a gapless scenario since then
the maximum of the susceptibility agrees better between both methods.
