why did my record player slow down, speed up, then go back to normal?

If it is a table with no fixed speed controller, it could have been a surge or power change in the system temporarily.  in other words, if a large load came on-line in your area, the frequency would changes for a very short time so that additional generation can catch up with the load.

So, if you have a 60 hz system, it is fluctuating between say 59.5 hz and 60.7 hz.  (just an example).  So, most people use a stroboscope disc and a regular incandescent bulb and adjust the turntable speed for exactly the correct speed.  But, this only works if the system frequency does not change and it always changes.  Electrical system controllers and auto transformers adjust voltages to compensate for  system load changes.

So, you could have experienced a system surge in your area for a short moment.

If you have a speed controller, that is different.   A speed controller takes the 60 Hz frequency (USA) and voltage and converts it to DC, the converts it back to a fixed line voltage and a fixed unchanging 60 Hz, making your turntable speed always fixed regardless of changes in system frequency from the wall.

Unless one has a speed controller, internal or external, the turntable speed is subject to changes based on the electrical system changes based on loading.

Of course, this is assuming that nothing went wrong with your system.

I hope this helps.  

For fun, get an inexpensive (a few dollars) stroboscope disc and a incandescent bulb and check the turntable speed.  

enjoy

There are also apps on line that provide for your cell smart phone to do this same check.

It is interesting.

Apps such  as RPM, Rotor or true note.

enjoy,

Power surge, or bad belt.

Tenet? 

Happened to me with my older Luxman PD-277.  There are speed control potentiometers for 45 and 33 RPM and they were showing their age.  A little DeOxit cleaned them up.  The wiper contact was a little dirty.