  We have identified a large-scale structure traced by galaxies at z=0.8,
within the Lockman Hole, by means of multi-object spectroscopic observations.
By using deep XMM images we have investigated the soft X-ray emission from the
Warm-Hot Intergalactic Medium (WHIM) expected to be associated with this
large-scale structure and we set a tight upper limit to its flux in the very
soft 0.2-0.4 keV band. The non-detection requires the WHIM at these redshifts
to be cooler than 0.1 keV. Combined with the WHIM emission detections at lower
redshift, our result indicates that the WHIM temperature is rapidly decreasing
with redshift, as expected in popular cosmological models.
