What brand and "class" SD memory card are you using? Likely the memory card isn't fast enough to keep up with the HD video. "Class 6" or above is recommended.
More extensive explanation here:
If your camera is not blinking a red non-circular icon, then it's probably the camera overheating, however it shouldn't be happening on a EOS 60D, so I believe it is a SD Card issue.
if your SD Card is Class 4 and below, DO NOT use it to take video, it's not possible. Get a SD Card as said above with Class 6 or above, [It is commonly symboled with a C wrapped around the #6 or #10], I recommend Class 10.
The Class represents write speed, so the higher the write speed, the better performing the SD Card is during a video or photo shoot.
As a reference, the symbol is like the ones in the green circle, make sure it is #6 or #10 in the C:
*These SD Cards in the image is outdated so don't buy one of these, buy the ones on store shelves.
As a personal recommendation:
I highly recommend a Sandisk Extreme (Class 10) SD Card with at least 30mb/s write speed (Cost for 16 GB or less is below $28, black friday deals could push a 16GB into the teens so look out for deals). I had bad experiences on Transcend and PNY Class 10 Cards (the Transcend was a waste due to it being all dead and lost a week worth of shooting; the PNY decided to not work with video for about a month).
09/26/2023: New firmware updates are available.
08/18/2023: Canon EOS R5 C training series is released.
07/31/2023: New firmware updates are available.
05/31/2023: New firmware updates are available.
05/18/2023: New firmware updates are available.
03/30/2023: New firmware updates are available.