Air source heat pumps are essentially the same thing as AC units, just fitted in reverse to heat the interior of buildings.
Theoretically you could fit the roofs of station platforms with a large number of AC evaporator units, there is plenty of space there to fit them. The problems would be the length and number (2 per unit) of the insulated refrigerant lines you would need to run to the surface as well as the space to put the compressor side units at street level or above on the stations. If enough cooling was installed it at every station it would eventually daisy chain the entire network, the tunnels would cool down through the piston effect from the trains which would also cool down as would the rest of the station through the cool air slowly pushing the hot out through the escalator shafts.
The other issue (and probably the one TfL won't admit is the real reason they won't do it) is the cost, the amount of AC kit needed to be installed to achieve this and the ongoing electricity to run it would be huge.
That is what has happened organically over the life of the tube, heat has seeped into the stations, tunnels and surrounding ground and this is what keeps the whole system permanently hot rather than the cold temperature that naturally exists below ground level which caused the tunnels to be cool year round as advertised in the early decades of operation.