I have a cold body of constant temperature A and a hot surface several times it's area. What is the best way I can absorb the heat from that surface using A. Will sheets of metal like alluminium or copper help?
A shape such as the one below will help. Make the piece from metal (copper). Insulate the edges.
As the differences in area get larger, this will eventually have diminishing returns in improvement. When the goal is high heat transfer, thinner is better. Of course, at some point, the thin piece will be the same as just putting the smaller cold piece directly on the larger hot piece. The trade off is between the heat transfer perpendicular versus lateral in the piece. When the goal is uniform temperature on the (larger) hot surface, thicker may be better. In the latter case, an optimal shape (curvature of the sidewalls) may be possible instead to minimize lateral temperature differences versus the perpendicular temperature difference.