I would always recommend cooling for 180 °F water. There are at least two good reasons. One, as you noted is the fact that hot water is a terrible lubricant. Cooling it will improve this. The second reason is probably more important. There is a film of this water between the seal faces. Assuming it is a single seal, the other side of the seal is at atmospheric pressure. At the water film cross the seal faces it will pick up more heat. So, when the water reaches the atmospheric side of the seal, it is at or above 212 °F. It will flash to steam. If it reaches this combination of temperature and pressure before it reaches the atmospheric side, it will flash between the faces. This will disturb the seal and cause it to fail. So, it is not good enough to suppress flashing in the seal chamber. You need to suppress flashing between the seal faces. And, the only way to do this is with cooling. Our standard way to address this is an API Plan 23 seal cooler.
I should also note, that the preferred o-ring for hot water is often Ethylene Propylene (EPR). EPR is rated for continuous service up to 300 °F. But, the o-rings are subjected to some of the heat generated by the seal. If the o-rings are running above 250 °F, they may experience reduced life. This is improved with cooling. If you change to a high temperature bellow seals to get away from the temperature limit, then you have a potential problem with pressure. Most typical bellow seals are only rated to 300 psi. I don't know what process you are working with. But most of our relevant applications are boiler feed water. And, the pressure is often much too high for bellows.
I consider water to be one of the most challenging fluids to seal with a mechanical seal. You need all the advantages you can get. Cooling helps in a number of ways.
Johnny Pellin