12V Battery Pack, Is it safe to use with a Mega?

I bought a 12V Super Rechargeable Battery Pack from Ebay, link is HERE. These batteries do hold to their names. I bought this battery pack about 3 weeks ago and since then I only had to charge it once. And it not that I don't use it, no, actually I use it all the time and it still puts out around 11 volts. However, I have noticed that if I use this battery to power my Arduino Mega or Uno for about 10 minutes, the voltage regulators on both get quite warm.

The battery that I have is the exact same one in the link, 12V @ 1800 mAh, but being that the regulators do get rather warm, I'm thinking maybe I should use a 1 M (1 W) resistor to reduce some of that current.

My question is, what is the maximum / safe amount of current the Mega or Uno can receive before any damage is done to the regulator?

My understanding is that power supplies don't "push" current so to speak, but rather that the equipment "pulls" what it needs. So you can have a battery the size of a house that can provide a zillion Ah, if it's the right voltage and the equipment needs 200mA or 20mA, that's all it'll draw.

E&OE, YMMV, Ts and Cs apply...

True, but the regulators shouldn't be getting as warm as they are, I can feel it through the back of the board. The only things that are running off the regulator is the LCD screen and accelerometer.

Maybe this is normal and I just never realized it until now.

Yeah... my next door neighbours are away, and I just went to feed the koi and check the swimming pool. So I just noticed a strange noise in their pool gazebo, where I see they have a power supply thing (no idea what it does) which sounds like a garbage disposal- probably the fan catching or fan motor bearing. Question is, has it made that noise all the time and I never noticed or is this a new thing.... Luckily they get back today so they can decide....

Up until now, I had the battery powering my Uno and didn't notice anything out of the ordinary because I never checked it. It wasn't until I started using the battery to power my controller (Arduino Mega) and actually hold it in my hands, did I start to feel the regulator get warm. Before then, I always powered my Mega via USB, but occasionally I used a 6V (bulky) battery pack for simple out door tests.

I'll try to get a small heat sink, to hopefully dissipate some of the heat.

I don't even know which chip is the PS on a Uno....