Yes, it's a 1st world problem (and one we'd not even thought about until tonight). When our builder cleverly suggested that he double-coiled the underfloor heating by the back door we thought it was a great idea...and then we plonked a big thick coir doormat on top of it...seven years ago!
Most of the time the floor is warm, but can get quite hot if we've left the back door open for a bit so I think that rules out anything with a rubber backing.
We need to walk along a muddy lane and garden to get to the back door so I guess we could put something the garden side of the threshold, but the pets and children don't know how to wipe their feet.
Is there some sort of slatted/brush combo available? It doesn't need to look pretty, because it is only cold for a couple of months here.