The heat is part of the price you pay for using a camera designed primarily for still images as a video camera.

Dedicated video cameras with sensors the size of most DSLRs cost about as much as a nice automobile or even the price of a small house and have much more cooling ability built in. So do some of the newer high end DSLRs such as the Canon 1D X Mark II that has a computer style heat tube running from the back of the sensor down to a heat sink in the vertical grip at the bottom of the camera.

When production companies started using DSLRs for some broadcast grade products, such as the network drama "House", they used multiple bodies so that each could be swapped out for a cooler one after shooting for about ten minutes or so. Any longer and the heat buildup around the sensor started to detrimentally affect the sensor noise levels and overall image quality.

Most cameras, including your 600D, have temperature sensors and firmware included in the design that monitor the vital statistics of the camera's internals and will shut down the camera before it is allowed to heat up to the point that lasting damage would occur.