A few points about film processing temperature: We control the activity of the developer, stop bath, fixing bath, and wash water, by adjusting chemical strength, time in solution and fluid temperature. In the era when most of this chemistry was established, 20⁰C (68⁰F) was typical room temperature. There is no magic in 20⁰C, you are free to process above or below. Most all modern color films are now processed at a temperature near 38⁰C (100⁰F). Charts are published cross-referencing time vs. temperate.
It is best if all fluids are adjusted to about the same temperature, this incudes wash water and washing aids. If any fluid should differ widely, reticulation may occur. This is a shattering or cracking of the film emulsion, at worst. At least, mild reticulation resembles elevated grain structure. This happens because films are sandwiches of coats. Each layer has differing rates of contraction and expansion with temperature. If films are plunged into differing temperatures, these layers may slide against each other and fissure.
Again, all fluids should be nearly the same temperature, including the running water wash. My experience passes along, measure the temperature of the running water and if possible adjust by mixing the hot and cold valves to the desired temperature.
For the home darkroom, a water bath will be the best temperature regulator. Procure a pan that will hold all the bottles or beakers of process including the developing tank. Pre-mix and measure out all solutions. Glass beakers are OK but stainless steel is best as metal beakers speed temperature equalization. Fill the pan with water and partially submerge the beakers up to about 75mm (3 inches) deep.
Use a thermometer, measure all including the water bath fluid. Add ice or hot water to the water bath to set it and keep it at the desired temperature. After the film has been loaded into the light proof developing tank, pour in the fluid; agitate continually for the first minute. Place tank in the water bath. Remove tank from water bath, agitate 5 seconds every 30 seconds and replace the tank in the water bath between agitations.