Amp fan cooling without dc current in the 110 line

Half wave rectification puts a little bit of DC on the AC line, but decent SMPS’s are full wave and a lot of muffin fans won’t even work right on half wave. A little 5V wall wart will power one just fine.
That said, putting an amp in a box that requires active cooling is a mistake. I’ve built computers with up to 6 120mm fans to keep things cool and you’re in there blowing out dust every 3 months in a religious sort of way. I’d never think of putting my F5 in a cabinet. I wouldn’t even put it on the second shelf of a rack.

I realize that an amp in a cabinet is not ideal. However having a kid click the power button on and off but repeatedly isn't great either. Really kids and HiFi don't mix. But I don't live in a perfect world so I work with what I've got. The ventilation in the cabinet isn't too bad, just need something to give it a little extra cool. So how do the batterys work, are they always charging and the fan runs off the batterys. Where can you get one.

The right way to do it is to use an exhaust fan, either out the back or out the top. Any way to open up the back of the cabinet?

I sympathize, even if my pretty unruly kids never did this. Ultimately, of course, this is the product of the gross inefficieny of many amplifiers. All that electricity that goes in does not come out as electricity at the speaker terminals, but a lot of it ends up as heat.
We are bound to see big improvements here as (at least EU) regulations will demand more efficiency, to meet our emission reduction ambition. Here in the EU this has already happened with many domestic appliances (see here for televisions: http://ec.europa.eu/energy/en/topics/energy-efficiency/energy-efficient-products/televisions) We recently bought a new vacuuum cleaner because the old one (a top tier brand) had died after 15 years. The old one was 1200 watt and the new one (conforming to new EU regulation) only 600 watt. The new one is lighter to carry, much (ie. really much) quieter, and it actually cleans better. I was recently looking at a new refrigerator to replace a 15 year old one, and the new models consume about a third of the old one. Ditto with the new office computers in my university.
This is bound to happen with audio equipment as well. Class D amplifiers are becoming more and more common. The only reason power hungry designs have not been outlawed yet is probably that there are only few of them.

Yea I could put an exhaust fan out the back panel at the top. There's holes in the bottom shelf underneath where the amp sits. It appears the amp vents on the bottom and the top. I'm hoping cooler air will draw in through the bottom up through the amp and I can exhaust it out the top back