Anemones will move around for a number of reasons, it may not be that it is seeking a different temp. It could be trying to get to a higher or lower flow area, could be seeking more or less light, or it could be that some are trying to get away from others. Anemones can be very aggressive towards each other if they feel at all crowded. They do not all get along, even the same species. How many anemones are in that 20L tank? More than 1 or 2 could be the overall issue, even if it's a result of them splitting. It may be time to move some of them to another tank.

Of course in my tank it would risk being poked with the handle of a toothbrush until it commits suicide by powerhead.

