I think loveracing may be on the right track here. Perhaps supersport always had a small booster vacuum leak, which gets closed off (frozen over) until the engine compartment warms up.. or there is a bit of ice forming within the booster, preventing it from doing its job. A vacuum booster not working properly can cause the symptoms the OP explained.... perhaps a frozen check valve. Have the pads ever been replaced? 77,000 miles on the rear seems overdue from what I've read here.
This is what the service manual suggests:
HARD PEDAL OR HIGH PEDAL EFFORT
A hard pedal or high pedal effort may be due to lining that is water soaked, contaminated, glazed, or badly worn. The power booster, check valve, check valve seal/grommet or vacuum leak could also cause a hard pedal or high pedal effort.
These are the test procedure from the service manual:
DIAGNOSIS AND TESTING - MASTER CYLINDER/POWER BOOSTER
1.Start the engine and check the booster vacuum hose connections. A hissing noise indicates a vacuum leak. Correct any vacuum leaks before proceeding.
2.Stop the engine and pump the brake pedal until all vacuum reserve in the booster is depleted.
3.Press and hold the brake pedal under light foot pressure. The pedal should hold firm. If the pedal falls away, there may be an external leak or the master cylinder is faulty (internal leakage).
4.Start the engine and note pedal action. It should fall away slightly under light foot pressure, then hold firm. If no pedal action is discernible, the power booster, vacuum supply or vacuum check valve is faulty, proceed to the POWER BOOSTER VACUUM TEST.
5.If the POWER BOOSTER VACUUM TEST passes, rebuild the booster vacuum reserve as follows: Release the brake pedal. Increase engine speed to 1500 RPM, close the throttle and immediately turn off the ignition to stop the engine.
6.Wait a minimum of 90 seconds and try brake action again. The booster should provide two or more vacuum assisted pedal applications. If the vacuum assist is not provided, the booster is faulty.
POWER BOOSTER VACUUM TEST
1.Connect a vacuum gauge to the booster check valve with a short length of hose and T-fitting.
2.Start and run the engine at curb idle for one minute.
3.Observe the vacuum supply. If the vacuum supply is less that 12 inches HG (406 millibars), repair the vacuum supply.
4.Clamp the hose shut between the intake vacuum source and the check valve.
5.Stop the engine and observe the vacuum gauge.
6.If the vacuum drops more than one inch HG (33 millibars) within 15 seconds, the booster diaphragm or check valve is faulty.
POWER BOOSTER CHECK VALVE TEST
1.Remove the power booster check valve from the power booster.
2.Using a hand operated vacuum pump, apply 15-20 inches HG (508-677 millibars) vacuum at the booster side of the check valve.
3.The vacuum should hold steady. If the gauge on the pump indicates vacuum loss, the check valve is faulty and should be replaced.
2011 Grand Cherokee Overland V8, 2009 Liberty Rocky Mt V6, 2000 Grand Cherokee Laredo I6, 1979 CJ7 I6 Quadratrac