It's plausible. But understand that one can imagine putting heat into underground masses with extremely long time constants, artificial geothermal in essence. It's possible that this wouldn't be practical but I'd want to see details. Geothermal heat pumps already do this.
It's true, there are district heating systems using subterranean seasonal sensible heat storage accessed through heat pumps, like Drake Landing. I wonder if the tighter temperature precision requirements of efficient adiabatic CAES would make it difficult — such a reservoir necessarily has enormous thermal mass, and if its temperature has to remain essentially constant during operation you must store vastly more energy in it than the "tidal volume" of heat during a yearly cycle. But maybe you can avoid such a requirement by clever enough engineering using several such reservoirs or a continuum of them (like packed-bed heat exchange media).
Constant temperature in such thermal stores is achieved by some flavor of counterflow heat exchange. One could, for example, send a fluid through a long bed of some granular material, in one direction for storing heat and in the opposite direction for withdrawing it. The temperature will be hot at the charging input end and cool at the discharging input end.
Yes, that's what I meant about packed-bed heat exchange media—but making the bed long of course gives it more surface area to conduct heat to the surrounding rock. And the moving thermal gradient inside the packed bed still gives rise to some temperature variation over time at the hot end and at the cool end. For district heating this doesn't matter.
Possibly this is less of a problem than I imagine it to be for adiabatic CAES; my knowledge of these systems is obviously pretty sketchy.
