Front A/C blows warm, rear is cold

well it is a blend, mode door (dampers) problem
it is a purely air distribution error.
you must check them all, to be sure they all work correctly
the service manual covers it.
https://eautorepair.net/Marketing/Default.asp

if you fix one problem reading this manual you just earned $80
so the manuals are free, ROI econ 101.

Usually a sign of low coolant or a blockage in the circulation.

How do you know it gets hot? Are you going by the temperature gauge on the dash? Or have you checked under the hood and found out the engine really is running hot? ...Reason I ask is, something just doesn't sound right in your description. After just 5 minutes from starting and driving away, the engine would not yet be at normal operating temperature. That would explain the heater not yet warm. But the temp. gauge says hot?
If so, I would replace the temperature sending unit on the engine. the one for the dash gauge, and see if that was the problem-not reading the temp. accurately.
Couple of other things: Are you sure you got all the air out of the coolant system? Start the car with the radiator cap off and let it run until the engine is warmed up fully, then shut it off and install the cap. Does your reservoir tank stay about half full where it should be, between the full-hot and full-cold marks?
If your problem persists. I would have the coolant system pressure tested, and see if you could get a true temperature reading of the coolant when it seems to be running hot.
Good luck.

often on the vans with rear air the air dor acuator goed bad and can no longer switch from hot to cold. also the swich itself will often go bad. this can be caused by sticky fingers damaging the inside of the switch. sometimes a good cleaning in warm water will fix a sticky switch

If this is equipped with a rear heater core it is probably plugged up and needs to be disassembled and cleaned out

I believe there is a expansion valve located on the back part of the air conditioner on the back right side i believe that could be the problem. I think if I am understanding you correctly, your back air is not as cool as it should be or not cold at all. Might want to check this. If I am not understanding you correctly please let me know again what your problem is.

there are two a/c lines that run down the fire wall under the van to the rear.right around the bottom of the fire wall the is a connection point on both lines. you will find a leak there .replace the orings and recharge system. also you will need two specail toyota tools to release the couplings.i have done this repair on several sienna's. let me know if this helps.

the blendor actuators strip out the gear inside the actuator cases on these all the time. replace the actuator.

I have the same problem with my 2001 Tahoe. The front blower is working, its blowing strong, but its cold air even with the temp setting as high as it can get. The rear blows hot air. The engine temp guage is at about 210ish which is close to being normal (220). I just replaced the thermostate a few months ago, I don't know if its bad again, or it I deluted the antifreeze too much and it frooze the heater core, or if the heater core just went bad. Any one know what the cause could be?

First thing I'd do is make sure your coolant level at radiator is up to proper level.Your auxiliary heat/cool unit should be in behind the right rear interior panel . (passenger side) If you remove that panel & check the hoses or lines going into it, they should be warm/hot to touch when heat on & engine warmed up to operating temperature. If not, it's most likely a problem with the electronic control, not allowing coolant flow to it, and I'm almost sure that the control cannot be fixed, but needs to be replaced. Triple check all your fuses for heater controls etc, but if everything else is working except the heat part, I'm guessing control at dash needs replacement.