[Slightly vulgar shower thought]
How does excretion (defection or urination) affect the thermal energy of the body? The substance retains some thermal energy from the body, but does losing that substance cool the body in any way?
But the piss will eventually leave your body, so all of the heat energy that went into it is going to be dumped out.
Well it starts at body temperature, and if it stays in thermal equilibrium, then it doesn’t require energy to maintain. It wouldn’t make a difference, you’d be losing the same amount of heat energy either way.
Based on what others have said, I think the answer is that it isn’t really in equilibrium. If the bladder is in contact with tissues that are warmer on one side than the other, then it will basically act as a thermal conductor to move heat away from your core.
Your body isn’t a closed system. You are constantly radiating heat away, and you are constantly burning calories to maintain your body’s core temperature. So the bladder is a tank full of mostly water inside that is draining those valuable calories and then will just be pissed away.
It’s not like the bladder is radiating heat away when it’s inside the body. Heat doesn’t just get used up, it has to go somewhere. If it was in equilibrium it would just increase your thermal mass and have no effect.
Like I said though, I think the issue is that it’s constantly moving heat away from the core to the lower abdomen where it can be radiated or conducted away.