Ummmm's answer might be short but it's almost certainly the best you'll get.
By October the whole of Europe can be rather cool. You MIGHT get some warm sunshine in southern Spain, or similar destinations, but there's little certainty of it. You really need to heading further south which, with just a week to spare, means Africa. Most mainland destinations in Africa are quite pricey but the Canary Islands usually have loads of cheap deals available. (The Canary Islands are politically European, since they come under Spain, but they're definitely African if you look at a map!).
Others suggestions which frequently occur here when someone asks about finding sunshine in October are Cyprus and Egypt. Cyprus is geographically in Asia (even though it's normally thought of as a 'European' destination), so the longish flight there is taking you mainly eastwards, rather than south towards the sunshine. Egypt is a better bet but holidays to there are usually much dearer than to the Canaries.
Chris