Why is ice cold water rather than room temperature or warm water used to wash the remainder of the benzoic acid out of the beaker?

As a general rule:

"Solubility of compounds decrease with the increasing of temperature" as you can see in the graph.

Also, filtration, is a chemical process where you are separating a precipitate from a solution.

Now, if you want to recover benzoic acid from an aqueous solution, the better option would be with Ice cold water because its solubility will be lower because temperature is lower producing more precipitate.

Learn more about this process in:

a saline water is fed to a reverse osmosis unit at flow rate o rotameter connected to reject (retentate) line reads 8 installed in reject and permeate line reads 15.2 milli siemens/cm and 300 micro siemens/com respectively.

the answer is barium and copper bromide