2004 toyota corolla idle jumps

All the above is normal , except 2000rpm is a little high at idle . can be adjusted down to 1500rpm. the computer adjusts the engine speed . in drive the transmission loads up the engine idle drops to 1000rpm . in park engine unloaded rpm increase all normal.

my guess, car unseen.
car has an A/T trans, and all A/T jump from park to drive
selection is made on a cold engine, due to the very fast idle speed.
what was the park and cold idle speed. and out door temps.
the lower out door temps, make idle speed , cold engine, even higher
but when the engine reaches 180F or more and holds, the RPM
is about 800 RPM (the FSM tells the exact amount and does vary for the unstated engine option) so at 800 the tranny don;t lurch, did it on a hot engine. so if idle speeds are wrong say so and say it with engine temps. or its all meaningless.
next :
runs rough, the engine (even parked?) is called misfire.
misfire on a good engine (full compression)
is:
1: bad spark (or never had a tune up ever) every 60k miles.
2: bad fueling. too rich, too lean (low idle) bad injectors (stuck or clogged, or wrong fuel pump pressure or bad Fuel pressure regulator)
3: bad fuel.

there is a test for each possible failure. all very simple test.
it takes $80 in gauge tools to find engine problems.

You have a vacuum leak Could be old vacuum line broken somewhere Brake booster vacuum line off Remove air cleaner and inspect all hoses

chech your pvc value first if that does not fix it it could need a new intake gas

IDLE SPEED COULD BE TOO LOW.IDLE SPEED SHOULD BE 800 RPM. IF IDLE SPEED TOO LOW AND ENGINE NOT WELL TUNE UP AIR CONDITIONER WILL PUT LOAD ON ENGINE CAUSE ENGINE OVER HEAT AND SHUT OFF.

not normal. bad spark plug wires they may be gapping or jumping spark. also maybe coil pack is doing same thing.

adjust the accelerator cable.

you need to replace the throttle position sensor on the intake.

Were there any water leaks into the interior? Or did you ever jump start someone else's car? I had the same exact problem resulting from jump starting a friend's car. It spiked the electronic control unit (ECU) computer. I changed the ECU and the problem was fixed.

start by pulling hose off egr valve see if idles smoother almost instantly