There are a few experiments comparing the reactivity of alkali metals with cold water. Is there any purpose to use cold water? Does the temperature of water affect the reactivity of alkali metals?
Increasing temperature does indeed increase the reactivity of alkali metals, or most things for that matter. The rate of reaction increases because the average kinetic energy of the molecules increases, and so more collisions can overcome the activation energy for the reaction to take place.
The reason for using cold water rather than hot water for the reactions of alkali metals is that alkali metals are already reactive enough at room temperature. Even fairly small lumps of potassium will burn if you put them in cold water so imagine what it would be like with hot water. This is not something you want to try without very careful precautions.
The rule of thumb is that each 10 Celsius rise in temperature doubles the reaction rate. So increasing the temperature certainly wouldn't slow the reaction. I doubt that the temperature rise would yield a very good correlation however to the overall reaction rate.
The metal is a solid so it is a different phase. The reaction is exothermic, so the temperature at the solution/metal interface is certainly higher than the temperature of the bulk solution. So the overall rate of reaction would be very complicated and depend on surface effects of the metal.