This is what a pristine board should look like:

Heat Load

It's logical to assume that most of the heat inside the XDR's case is coming from the power supply. But further investigation suggests this is incorrect. If you remove the power supply and isolate it from the rest of the stuff crammed inside that small case, the power supply runs very cool under the XDR's load.

The tuner/DSP & HD Radio modules inside the case get very warm on their own after just a few minutes. Another contributing factor to heat is all the precision voltage regulators on the main board, most of which are on the bottom.

My conclusion is the case as it's designed just can't adequately dissipate the combined heat load from everything.

Experimental, not offered as an upgrade: Transplanting the Sony's power supply to an external case reduces heat in the tuner itself. I didn't expect this to work well without a vented case for the power supply and possibly its own fan. Surprisingly, the power supply runs very cool, even in a completely enclosed aluminum case with no venting at all. At an ambient temperature of 29C, the top of the case levels out at about 35C. The maximum measured internal component temperature is about 45C. In a cooler ambient environment of about 22C, the external and internal temps are correspondingly lower. These measured temperatures are all perfectly acceptable.

But The XDR Still Absolutely Needs At Least One Cooling Fan!

Even with the experimental external power supply conversion, the XDR's internal components get very warm. There's no getting around the need for a cooling fan to keep everything as cool as possible, especially those pesky modules that are no longer in production and very expensive to replace.

Notes About the XDR's Unregulated Linear Power Supply

NOTE: Don't poke around randomly inside the XDR while it's plugged in to AC power. There's live exposed 120VAC inside the case!

The XDR's power supply design is very simple. The AC transformer has two secondary outputs that are rectified and produce unregulated +10.5VDC and +5.2VDC. Since they're unregulated, they naturally vary up and down depending on the load on the power supply. And even small variations in the AC line voltage will cause the power supply's unregulated outputs to change accordingly.

Being unregulated, the power supply naturally has quite a bit of ripple under full load. It looks like a sawtooth on an oscilloscope. My power supply upgrades substantially reduce the ripple.

With the tuner in Standby (Off), the nominal +10.5V output rises to about +12.5V. The nominal +5.2V output rises to about +7.25V. This is completely normal. Once the tuner is powered up and stabilized, those voltages will fall to their nominal 10.5V and 5.2V shown in the XDR's service manual. However they can drift as low as about 9.9V and 4.9V respectively and the tuner will usually still run fine because all the precision voltage regulators can still produce the necessary regulated voltages. (But if the voltages are low due to capacitor failure, that's a different issue)

UPDATED 3/3/2021 IC 902, +8.5V Regulator: I suspect Sony engineers deliberately designed the +8.5V regulator circuit to actually produce ~8.25V: Sony knew the main power supply's unregulated 10.5V output could drop quite low along with a drop in line Voltage. That could in turn cause the 8.5V regulator to drop out. So they deliberately designed the 8.5V output to actually be 8.25V, making it more tolerant of low line Voltage. That's my theory anyway. There are other write-ups about the XDR that speculate the +8.25V they measured in a single sample of an XDR was a fluke. It's not a fluke - It's about 8.25V in every XDR. If you run the math on the voltage regulator control circuit, the calculated output is about 8.25V. Installing a 32.4K resistor (or something close to that) between IC 902 Pin 4 and Ground will raise the output of IC 902 to about 8.4V-8.5V which is much closer to spec. But you would probably only do that if you also beef up the +10.5V side of the unregulated power supply so it doesn't droop too low.

The photo below shows the long-term effect on the XDR's main board from running the tuner continuously without fans. The board is literally heat-scorched around voltage regulators (most of them are on the bottom solder side). It even smells burned.

XDR Heat Problems: Give it as much ventilation as possible!

Internal component operating temperatures without a fan, and with fans, are shown below.

Typical symptoms of XDR heat damage include:

Tuner gets stuck on flashing WAIT message
No signal strength or audio from any station you try to tune
Persistent hum in the audio
Choppy "helicopter noise" audio when tuned to an HD station

Sometimes the power supply isn't delivering adequate voltage or current to the main board because the main aluminum electrolytic filter capacitors in the power supply have severely degraded. The electrolytic capacitors on the main board and inside the tuner/DSP module can also degrade to the point where the tuner doesn't sound good or isn't working at all.

The HD Module is very susceptible to heat-induced failure. If you get choppy audio when tuned to an HD station, your HD module has probably failed. The only source for a replacement module is removing a working one from another Sony HD tuner or radio that used the same module.

I'm also seeing increasing numbers of XDRs with long-term heat damage affecting the plastic case. The most common point of failure is at the mounting posts for the transformer. That transformer gets VERY hot. Eventually the plastic screw mounts for the xfmr become brittle and crack. They literally fall apart. Here's an example with both posts broken off.