If you want to make a hot compost system I'd get a sheet of kingspan insulation (other brands are available). Stick this against the the wall this will help to keep the heat in and also it doesn't absorb water so is a very good barrier to prevent moisture contact against the the bricks. On the lid or no lid I cover my heap's with plastic, old compost bags are great, then cover with several layers of carpet. The plastic retains moisture (hot compost steams and loses loads of moisture) and the carpet keeps it all cosy. Unlike @posy I go for hot fast compost and loads of it
I don’t cover a compost heap while I’m ‘making’ it ... adding stuff, building up the layers etc ... but once it’s as big as I want it and I’m ready to start building another, it’s then that I cover it and leave it to ‘cook’.
Gardening in Central Norfolk on improved gritty moraine over chalk ... free-draining.
I wrapped my three 'Dalek' bins with old fleece/bubblewrap and have been pleasantly surprised at the difference it makes, there are lots of brandling worms still in there which is good. Not so keen on the look of the bins though so have covered them with a roll of old brushwood fencing. Recycled bamboo blinds from Recycle also make good cover ups I find.
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Gardening in Central Norfolk on improved gritty moraine over chalk ... free-draining.