That's not an issue for a device like this. The typical use case is refreshing the screen once every 5, 10, 15, etc. minutes, which takes a few seconds (including connecting to wifi, downloading the data, etc.), and then spend the rest of the time in some super low power deep sleep mode, drawing maybe tens of microamps. Or maybe checking over Bluetooth LE whether to trigger an update, once every couple seconds. This will never get hot enough to the point where it would matter.
OP said that they did this with a first-gen kindle, and I would not be surprised if it had an always-on wifi connection and was processing data much more frequently then you suggest. I agree that there are plenty of ways to build this in a power-efficient way, I just want to suggest that given modern battery controllers, heat is a much more likely cause of failure then keeping the kindle plugged in all the time.
