I have two theories. [edit: theories why anyone would come up with such an idea in the first place, that is]
First, E = energy, and temperature is energy. So if temperature increases, doesn't that increase E? And if E = mc², doesn't that mean that either mass or the speed of light would need to speed up in order to keep up with it?
Second, although false, a lot of people are trained to believe that time stands still at 0 K. In that case, light could never escape 0 K, and as temperatures approach 0 K light would slog to a halt. If that was the case, the logical conclusion would be that speed of light would increase as temperatures rise.
Or maybe something completely different - I just thought it was a fun question to try to reverse engineer. :)
Hypotheses, not theories. Unless your idea has been experimentally duplicated & verified by a good (if not great) number of your peers, it is not a theory.
Again, I'm not trying to be an asshole, but it's important to remember in science the distinction between the scientific meaning of the word as opposed to the widespread colloquial meaning of it.
No it is not. Temperature does not depend on mass, while energy does depend on mass.
If you apply thermal energy to two identical objects of different mass equally, the temperature will not be the same between them, as the object with less mass will have a higher temperature, despite the same amount of thermal energy transfer.
Temperature itself does not affect the speed of light - remember that space is freezing cold, and light moves through it just fine. So warmer temperatures don't do anything with time.
If earth suddenly gained a bunch of mass, that would change things up as gravity would increase. However, we wouldn't really notice, as everything would speed up more or less the same. We'd have to compare ourselves to someone in a system where time moves differently in order to notice.
Also it wouldn't matter to day to day life because time is based on your frame of reference, and you'd be inside the frame of reference so it would feel exactly the same to you.
Think about what it means for "time to move faster". If you're within the faster timestream, you may not even notice, unless you ascribe to one of the more interesting metaphysics which theorize that consciousness can exist outside of time. But if you're within a different timestream, you may notice the affect timestream speed up or slow down relative to your own timestream.
Now, I've read your potential reasons why temperature may affect the passage of time. As to E=mc², the energy is not increasing, but more potential chemical energy is being turned into ambient heat, eg. by burning fossil fuels. Super simplified, ofc. but it's not a case of energy being added to a closed system.