Yes. Water (as well as anything else) has a property called “heat capacity”, that is, how much heat is required to heat it up of, say, one degree Celsius. In fact, water is so important that one unit of measure for heat, the calorie, is calibrated exactly on its heat capacity: 1 calorie is enough heat to warm up one gram of water of one degree Celsius (the ‘calories’ mentioned in food and stuff are actually KILO calories, or Calories, or great calories, and it’s enough heat to warm up one kilogram of water… etc. etc. It’s confusing, I know: units of measure often are).
Heat capacity changes a bit with temperature, but not much. So, if you give, say, one calorie per second to one gram of water, it will heat up at a rate of one degree per second, and if you take one calorie per second from it, it will cool down at the same speed. Which makes a lot of sense! If for example, instead, water cooled down faster than it heats up, you could heat up one gram water from 0 to 100 degrees using 100 calories, then cool it down and get, say, 120 calories out of it. That’s 20 free calories out of nowhere! Of course it does not make sense – you can’t create energy like this: in physics, just like in that old song from New Radicals, “you get what you give”.
It will depend upon the way that the water is heated and cooled. Simone is correct in saying that you are limited by the total energy in and out but the rate of heating and cooling can be completely different and water’s heat capacity does change with temperature (though slightly).
Taking Simone’s example:
You can heat 1g of water up from 0’C to 100’C at using 100 calories of energy but you could this at 1 calorie per second (taking 100 seconds) or 100,000 calories per second (taking 1/1000th of a second).
On the other hand (when cooling the system is considered ‘open’) the water will cool at a rate dependent upon both the contact area and temperature of it’s surroundings. e.g. (assuming the cups are the same shape and originally the same temperature) a large cup of tea put into a fridge will cool quicker than a small cup of tea left on a table.
Heating and cooling rates can be completely different but the total energy change at the end is the same because of conservation of energy.