Maybe some translucent plastic on top? That MDF will fall apart in no time, I think, and some sunlight in there to warm things up ought to help with the composting. Yours probably will work just fine though since it's probably stuff you had laying around :)
I was actually thinking of covering the whole thing in black plastic to really get it going(and because I already have some :D). It's in the shade for half of the day anyway so I don't think that it would get too hot.