What is the place called where cheese is aged?

I don't think there's a specific name for it.

I found a book that simply refers to it as an "aging space/room".

The Aging Room: The "Big Three" Design Issues

When designing an aging space, there are three main factors to consider: temperature, humidity, and air (both air exchange and movement). Each of these factors will be influenced by the volume and type of cheese being aged; your area's native influences (weather and soil/earth stability and properties; also if your building is underground); and the building structure and size.
_{The Small-Scale Cheese Business: The Complete Guide to Running a Successful Farmstead Creamery p. 126}

Around the web, I've seen it referred to as a "cheese cave", which refers to the historical habit of using caves for aging cheese.

1. How to Make a Cheese “Cave” - This is a guide on aging cheese in a refrigerator.
2. Crown Finish Caves - A cheese aging facility in Brooklyn, NY.
3. Wegman's Food Market opened a "cheese cave" in 2014.

For a similar reason, I've seen a smaller number of references to "cheese cellar".