I'm running an UNO R3 off a regulated 9V supply (tested at 9V w/Fluke meter).
The back of the board where the regulator mounts is holding around 125F using a digital thermometer.
There's a shield on top so I can't measure the regulator temp itself. The board is otherwise in open air.
Is this too hot? Will it reduce the lifespan of the UNO?
You are probably stressing the regulator, I suspect by powering too many external devices. There is never any need to do that.
What are you trying to power from the regulator output and how is it connected?
What's that in real money? Sounds like a normal daytime temperature in some parts of the world!
Keep below 100C if you can.
Rule of thumb.
If you can keep your thumb on the hottest component without pain, your good.
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That's probably okay, but you're putting some stress on the regulator, and I'd try to make it a little better if I was going to continue using it...
If you're not going to be pulling more power from it than you are now, you can "quick fix it" with thermal epoxy (like arctic alumina or arctic silver epoxy) and a small heatsink (available dirt cheap on ebay).
If you can, lower the input voltage so the regulator doesn't have to drop as much voltage (the heat will scale down with the difference between input voltage and output voltage). At 7v input, it would generate half the heat that it does at 9v.
You may need to look into external 5v supply, or separate supply for some of the connected devices, etc.
If you can smell burning, its definitely too hot! Most components can be well above thumb
temperature and survive (but you then have to worry about long term roasting, heat dissipation
if inside a box, lots of things its best not to have to worry about).
The internal regulator of the Arduino is simply not suitable for powering more complex assemblies. Using a "shield" should be a hint that you need a proper power supply, a switchmode regulator powering the 5 V connection.
Powering the UNO Vin from a regulated supply does not make sense. (It would make somewhat more sense if you set the supply to 7.5 V). If you have a regulated supply, you power the 5 V line directly.
I'm basically powering the UNO, plus a SeeedStudio V1.1 Ethernet Shield, plus at most another 20ma of very intermittant load (LEDs that might light for a few seconds during an hour's period).
My existing 9V supply is fixed, so I can't change it. In the mean time, I inserted 3 1N4001 diodes in series with the 9V supply to drop the incoming voltage a little. The board is now running at 100F, but the diodes do get warm (not hot). Expected since they're each dropping around 0.6V at a several hundred ma.
Give what everyone has said, I think my long term answer is to rig up an external 7805 regulator w/heat sink and not overheat the UNO itself.
If I run the 7805 at under 1A, and assuming I have a heat sink to dissipate the 2-4 Watts that the 7805 will throw off, do I need anything else (filter caps) between the 7805 and the 5V IN for the UNO?
Thanks.
I would use a DC to DC SMPS rather than a 7805.
Lots in eBay.
do I need anything else (filter caps) between the 7805
Yes, filter caps. Follow the data sheet recommendations.
LarryD:
I would use a DC to DC SMPS rather than a 7805.
Lots in eBay.
+1. You can often get a 5V 2A phone charger at the dollar store.
+1. You can often get a 5V 2A phone charger at the dollar store.
I've thought about that. Certainly enough current. Just how well regulated and clean is the 5V coming out of these adapters?
They work great in all my digital and analog projects.
.
aarg:
+1. You can often get a 5V 2A phone charger at the dollar store.
Considering the "heat" on the cheap Chines ones, I prefer to get them branded at Garage Sales or "thrift" stores. 
Paul__B:
Considering the "heat" on the cheap Chines ones, I prefer to get them branded at Garage Sales or "thrift" stores.
Well, most of the ones I have were actually obtained at a surplus store. Some of the others are Samsung branded. I know they're good because I used them to power things like a PCduino that really cook.