This outtake from a product development engineer on Reddit states that the answer varies from product to product. I guess only the manufacturer knows for sure which approach would be best for each device.
Product Development Engineer here - it varies by product. It's a very interesting question, because thermal cycling (on/off) is a very stressful condition for electronics that causes a lot of catastrophic failures. On the other hand most devices are rated in hours useful life, so leaving a device on is generally not the best choice if it will spend a lot of hours without anyone using it.
For lifetime rating, products are generally subjected to three main categories of abuse (others if it will be an outdoor or waterproof product): Elevated temp. operating tests, thermal cycling tests, and vibration testing. Generally the weak point of any given product is due to a particular design choice, and thus could be any one of those tests. If you're trying to decide about how best to use a particular device, you probably need to look up how it tests. I've been looking for test standards for you to read, but none are free. In LED lighting, we adhere to IES LM-80, which describes mostly elevated temperature testing. For UL compliance we'll need to pass some ASTM vibration testing (maybe D3580), and for customer certification we'll need to have data on storage life and cycling.
I know it seems like a cop out, but the best way to ensure long life of your device is to operate it in a cool environment where it won't get any extra thermal stress, and to use it as infrequently as possible. The maximum stress from cycling would be if you turned it on just long enough to get hot, then left it off just long enough to get cool before turning it back on, so if you're turning on a TV 20 times a day, you're probably doing it wrong.