In reply to goatee: There was a lot of snow in June but it is very unusual. Not saying that the unusual won't happen again but normally from end of May until late September you are supposed to find a clear path (no crampons required) at 2000m. But I wouldn't go before May.
I am speaking for the western Pyrénées, surely someone will have more detailed advice for the centre/east.
For the time of the year, it depends what you are looking for. Are you interested in fauna and flora for example ? Then try mid June to mid July, best moment to see the pyrenean flowers and most of the animals.
If you are doing the haute route, I don't think you will suffer from the hot temperature (but maybe I am used to it).
If you don't like crowds, try to avoid mid August.