but I’d like to know what’s causing this.
Tesserax gave you his best guess.
if there is a "run-away" process taking up the base station's CPU resources. Having file sharing enabled, would be one of these processes.
As a person who knows a lot about network and routers I think he is on the right track. Just like your computer.. if you run google chrome at some point the CPU activity will go to 100% and the fan will start roaring.
The CPU in the AE is a dual core 1ghz.. and will get really warm if a process goes messy and starts running 100% CPU.
BUT there are other possibles like power consumption and the sensor getting false readings.
If you use iOS version or earlier airport utility before Sierra you can look yourself at the fan and sensor readings.. although they are not in clear.. simple to understand .. but here are mine as way to compare.
Open the setup on your AE.. and go here.
As something useful this was removed from Sierra I heard.. Airport Utility should be as toy like as possible before Apple will be satisfied.
You will see a stack of info in various logs
Open ThermalInfo .. last one on the bottom of the list.
I cannot easily translate those numbers for you.. The Tmps Internal are the key ones..
But I do have access most don't so I can tell you the real numbers so this might be something that helps.
envstat
Current CritMax WarnMax WarnMin CritMin Unit
[accel0]
Accl X-Axis: 1049 0 0 0 0 none
Accl Y-Axis: 47 0 0 0 0 none
Accl Z-Axis: -7 0 0 0 0 none
Accl Temp: 19 0 0 0 0 none
[applfan0]
Fan_rpm: 1750 RPM
T_hdd: 0.000 degC
Fan_dcy: 57 0 0 0 0 none
[emc141x0]
T_internal: 40.875 degC
T_external1: 38.625 degC
T_external2: 45.000 degC
T_external3: 46.500 degC
As you can see the 4 internal temps are from 38-46
Inside house temp is around 15C so the temp are 30deg above ambient..