I don't understand why it's A/C, what's the purpose of the slash? Are there other examples of the same usage?
I think it's to make a difference between AC as Air Conditioning and A/C as Air Conditioner. However, in India, we use AC for everything!
So, when you purchase a unit, you may not prefer to say that you bought an AC. Please note that when you turn on/off, both are okay as you can do that for air conditioning and conditioner as well!
However, I'd say there's no major difference left today. AC and A/C both are in practice (at least wherever I've read) and they are interchangeable.