Run some refrigerant behind the walls and use it to heat nearby buildings in the winter.
Technically that's feasible using heat pumps, and you're then using the tube walls as a ground heat store. Unfortunately the fluid and heat complexities of doing this are considerable just in terms of planning the heat loops, making sure the system can heat the buildings properly, etc. And I've some considerable experience in heat networks, and I'm confident that it would never be economically feasible. If you cool the tube walls, you need to move that to a long term heat store (logically a patch of ground a considerable distance away, up or down), then you need to take that out in winter via another heat pump. If the heat abstraction exceeds the heat input, you permanently chill the ground - get that wrong and the recovery time to be useable can be measured in decades, meaning the system is economically a total writeoff, go the other way and the heat pumps cannot cool the tube walls.
There is however a simpler approach that might work, and that is to use the piston effect to clear air through the underground system. You might think this already happens, with the gale of an approaching tube train, but in practice when a train pushes air into the station, it merely "short circuits" to a train travelling the opposite way, so very little air actually leaves the system. A "brush seal" half way between stations would work wonders to enhance air flow without exacerbating the gale effect on the platform, but they'd need to make station modifications to stop the air recirculating. This isn't actually that hard if there's the sort of access control doors seen eg at Westminster station, which could be easily modified to seal off the tunnel from the platform. The pressure levels aren't that great, the seal doesn't need to be anywhere near airtight, and they could make better use of this to clear warm air through the existing ventilation shafts - potentially even use entire tunnel diaphragms just behind existing ventilation shafts that move out of the way to allow a train to pass (and lightweight enough not to cause a problem if they don't!). Alternatively have powerful fans running against the flow of train traffic rather than diaphragms.
Personally, I think that could work, wouldn't cost much to operate, but I suspect TfL have put tube cooling firmly in the "too difficult for us" tray, and the fluid mechanics involved would be very complex - my Heath Robinson description would need some serious modelling. Now there's a job for machine learning.