If the run is made of mesh panels, it shouldn't get hot as long as there is a good amount of covering to provide shade as the sun moves. A tarpaulin will do the job. You can get different colours eg green camouflage.
I use this on the runs:
http://www.wickes.co.uk/Onduline-Gre...000mm/p/240059
It can be screwed down on a wooden framed run, or tie with rope on wire runs. On one run, I have a loose sheet to pull across the opening lid section (weighted with bricks on long planks in windy weather), but it could be cut to fit and screwed down (use the special plastic fittings or large washers to stop the screws tearing out).
It keeps the rain out and provides shade - but grass won't grow under that amount of permanent shade.
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