When temperature changes, relative humidity changes. Imagine I have a closed room at a certain temperature; then, I blow water vapor at a similar temperature. Can a change in relative humidity induce a (small) change in temperature? If it does, would the temperature raise or decrease? I have tried to find information but all I find is about the influence of temperature on relative humidity, not the other way round.
The point is that air at different temperatures can hold different amounts of water (the hotter, the more water). Therefore, the relative humidity decreases when heating up air and increases when cooling it. The absolute humidity, however, stays constant in an isolated setting, because water molecules are not leaving or entering the system.
Now to your question. If the added water is already gaseous and at the same temperature, $T$ will stay constant, because the mean kinetic energy per particle is the same as before. If the water you add is in liquid form like droplets, the temperature would decrease, because some of the kinetic energy of the air is needed to evaporate the water, decreasing the mean kinetic energy.
All this is assuming the relative humidity never reaches 100%.
If you add water as a vapor, the temperature won't change until your relative humidity reaches 100%. After that water will start condensing. This releases energy and the temperature will slightly increase.
If you add water as a mist of droplets, they will vaporize. This takes energy and the temperature will slightly decrease, but only until the relative humidity reaches 100%. After that evaporation will stop and the temperature will stay the same.
This answer assumes that you keep the temperature of water equal to the changing temperature in the room, as stated in your question.