physics --heat

Heat flows because of the difference in temperature, NOT because of difference in internal energy. Hence, a temperature equilibrium would not cause any heat flow.
Take, for example, a large glass jar containing water both are kept at room temperature. The internal energy of water is higher than that of the glass jar because there are more number of water molecules (hence, mass) than that of the glass jar. There is still no heat flow between the two, because they are at the same (room) temperature.
On the other hand, if the water is just taken out from the fridge and its temperature is lower than room temperature, temperature of the glass jar (at room temperature) is then higher than that of the water. Heat flows from the jar to water until the temperatures of both are the same.