That cooling light will normally stay on constantly during the cycle. It will begin blinking at the end of the cycle when the cycle goes into the cool down. If that light begins blinking shortly after starting the dryer in an automatic moisture sensing cycle then you could have a problem with the moisture sensing circuit. In that situation, see if the dryer will heat and run properly in a Time Dry cycle. If it works properly in the Time Dry cycle then I recommend that you check the sensor bars to see if they are coated with residue or lint that could prevent the control from detecting moisture. Those bars are on the lint screen housing just inside the dryer door. If necessary, you can clean those bars with rubbing alcohol.
If those bars are clean, then you could have a wiring failure or a control failure that is causing this problem.
If the cooling light stays on constantly and does not blink before the dryer turns off, then the moisture sensing bars are probably not causing this failure. You may see an error code when the dryer stops. I recommend that you check that display.