Can thermal energy be cool when its hot?
Yes. But a cooler body has less energy than a hotter object
There is no opposite of thermal energy. Thermal energy is energy that comes from heat, and therefore comparable to temperature. There is no "opposite of temperature," and there is no "opposite of thermal energy." If an object has high thermal energy, it is hot. The opposite of that would be having low thermal energy, or being cold.
Thermal energy is useful energy. However, thermal energy is usually just heat, and not very hot heat at that. And heat, unless hot enough to boil water is a bit difficult to move and to convert into other kinds of energy. You can't run an electric motor directly off thermal energy for instance. But if you live somewhere where homes needs to be heated as opposed to cooled then thermal energy most certainly can come…
you are asking 'hot' teacup vs 'warm' kettle. If the universe were 'warm', it would have more thermal energy than a hot' teacup, regardless of the temperature - thermal energy - of the teacup, or it would not exist from 'burning' up. the kettle and the teacup sizes must be defined before the thermal energy can be determined, as well as the specific temperatures involved. doveshawk.
In this context, radiation refers to thermal radiation. Nearly everything in the universe radiates thermally in the form of infrared "light". When something is hotter, it radiates more. All of this radiation has some energy associated with it, which comes from the thermal energy in the object. In the case of a hot drink, more energy is leaving the drink through radiation than is being absorbed.
Assuming that we are talking about the same container for both the hot and the cold water, and that the water is the same water, the hot water will have more thermal energy. If we are talking about cold water with something disolved in it and hot water that is pure, it is possible for the cold water to have more thermal energy, but this would depend on what is dissoved and the precise difference…