  It is well established that in the low-temperature limit, the two-dimensional
quantum Heisenberg antiferromagnet on a square lattice (2DQHAFSL) exhibits an
anomaly in its spectrum at short-wavelengths on the zone-boundary. In the
vicinity of the $(\pi,0)$ point the pole in the one-magnon response exhibits a
downward dispersion, is heavily damped and attenuated, giving way to an
isotropic continuum of excitations extending to high energies. The origin of
the anomaly and the presence of the continuum are of current theoretical
interest, with suggestions focused around the idea that the latter evidences
the existence of spinons in a two-dimensional system. Here we present the
results of neutron inelastic scattering experiments and Quantum Monte Carlo
calculations on the metallo-organic compound Cu(DCOO)$_2\cdot 4$D$_2$O (CFTD),
an excellent physical realisation of the 2DQHAFSL, designed to investigate how
the anomaly at $(\pi,0)$ evolves up to finite temperatures $T/J\sim2/3$. Our
data reveal that on warming the anomaly survives the loss of long-range,
three-dimensional order, and is thus a robust feature of the two-dimensional
system. With further increase of temperature the zone-boundary response
gradually softens and broadens, washing out the $(\pi,0)$ anomaly. This is
confirmed by a comparison of our data with the results of finite-temperature
Quantum Monte Carlo simulations where the two are found to be in good accord.
At lower energies, in the vicinity of the antiferromagnetic zone centre, there
was no significant softening of the magnetic excitations over the range of
temperatures investigated.
