Flexible & thermally conductive materials?
There is a reasonable answer that does not involve aluminium. Thin silicone sheets are extremely available, but an even better solution to your problem would be a polyurethane elastomer with a thermal conductive filler however these are less readily available. Obviously they are not going to be as thermally conductive as metals but that is the sacrifice you make (currently) for flexibility. 

Hopefully in the future graphene will be cheap and readily available but with the green energy crisis, I doubt it!