Cabinet Cooling

https://www.amazon.com/AC-Infinity-AIRPLATE-Thermostat-Cabinets/dp/B00QFWWZQO

I do not know about the noise produced

I have the same problem with my setup and solved it with a small fan clipped to the back of my media unit, directing the air towards the rear of my integrated. I'm assuming the back of your cabinet is open, like mine.

I can hear it between music cuts as it's a cheap fan and it's temporary until I get something like the small, Vornado desk fan, which is pretty much whisper quiet.
 https://www.bestbuy.com/site/vornado-flippi-v6-personal-fan-vintage-white/5064607.p?skuId=5064607

I've tried listening to it and it's so much quieter than the cheap fan I now use. It's just that I'm lazy and haven't gotten around to getting it yet, but with summer on the horizon, I'll be getting one soon. It's easy to position so it's basically a set and forget option. 

As long as the back of the unit is directly in the path of the air from the fan, it'll work like you want. Even my cheap fan leaves my integrated feeling like it was never turned on.

All the best,
Nonoise

Another option, with speed control. 10dB at lowest speed

https://www.amazon.com/Receiver-amplifier-cooling-thermoswitch-multi-speed/dp/B0106LT6B6

I have this and on level two its inaudible at my seat, very nice thermostat control , very well made.

https://www.amazon.com/AC-Infinity-Rear-Exhaust-Receivers-Components/dp/B01LW1Q429/ref=sr_1_8?crid=Z...

One can also go for AC if looking to have comfortable temperature.