Does the material a bed is made out of affect the cooling time of a part?
What bed material cools faster?

I found an extensive list which relates various materials to their thermal conductivity, k [W/mK]; the lower thermal conductivity, the better the material insulates, and the slower the print bed will resist changes in temperature - both heating up, and cooling down. 

Here are the thermal conductivity for some common materials for 3d printer beds:

Aluminum    205
Glass       1.05
Acrylic     0.2
Air         0.024 (for reference)


There is also the matter of thermal capacity, but I will not go into that right now (need to do some research myself first!).

Will bed material affect cooling time?

Bed material, I believe, is not necessarily related to print cooldown time: it depends on the situation, such as whether we are discussing cooldown during or after printing, and if the bed is heated or not. 


If you are not using a heated bed, I believe the bed material doesn't matter at all.
With a heated bed while printing, only the first dozen layers or so are probably affected by the rising heat sufficiently that it affects the printing process.
With a heated bed after printing, the thermal characteristics of the bed will determine how quickly the print cools (and thus can be removed).


Also remember that other physical properties, such as flatness (both cold and during heating) of the bed material is vital for successful prints, and that not all materials can tolerate heating equally well!