The general assumption that I've found helpful is lukewarm/warm/hot to preference during and while soaping off, which opens the pores to cleansing. After cleansing, cooler or lukewarm (not cold) to close the pores afterwards. That sequence also happens to be the most refreshing from my experience, aside from any minor advantages.
Other than cold water, which is jarring to the system, I don't think the temperature of the water makes any difference to the body or to the pump, go with personal preference.