However, if the buildings above were to sink ground source heat pump loops into the warmed ground to heat the buildings in winter, this would basically be what you just suggested, and would be a win-win situation.
500 of them could remove 1.4 GW of heat.
Of course there are many ways to improve efficiency, but even assuming the worst case it’s still technically feasible to remove many times more heat than the line generates.
GP is saying that you can approximate the energy going into the system by looking at the electricity consumption of the trains, as all then energy is eventually going to end up as heat.
A heat pump can have a CoP topping out at 5. So 1 unit of energy needed to move 5 units heat out. That means a “net zero” cooling system would consume a minimum of 20% as much energy as the trains themselves. Realistically it’s probably closer to a CoP of 3.5, so 28% more energy. For something like the underground that gonna be a 5-10% increase in there operational costs at a minimum. Where does the funding for all that come from? And that before we even look at the capital costs of heat pumps and various ancillary equipment needed to run them.
As a point of reference TfL underground trains have an average power consumption of 140MW continuous. Now only about 45% of the underground is actually underground, but that’s still 63MW in just the underground parts. At an optimistic CoP of 5, that means 12.6MW of additional energy needed to cool the tunnels using your approach.
Wholesale electricity prices in the UK are something like 7p per kWh. So over a year that’s an additional £17m of electricity, just for cooling.
To put this in perspective, Sadiq Khan already runs the system at a massive loss. They just about cover their operating costs at the momemt, and rely entirely on grants from the rest of the country to do upgrades of any kind. So cooling efforts have to be very cost efficient. Also the UK grid is very supply constrained. New nukes are being built but there have been the expected massive cost overruns and problems, so any large new energy demands in the UK just aren't happening anytime soon. It's actually been deindustrializing due to very high electricity costs.
Also the underground is funded pretty much entirely by fares paid. Past UK governments have cut any tax payer subsidies for TfL to zero for day-to-day operational costs, and there’s zero indication that’s going to change any time soon.
I didn’t claim all technically possible solutions would be politically acceptable.