This is an interesting question and I will take the bait.
Using the definition from here, and considering 2000 grams (2l) of water at about 0 degrees Celsius which the body must heat to about 37 degrees Celsius the lost heat effort is not that big: 37 deg * 2000 g = 74Kcal (out of about 2000K calorie per day)
This article dives into the results of a study that deals with this exact question:
Sure enough, unlike previous studies, the new study found that
drinking hot water triggered a sweat response that more than
compensated for the heat of the drink. Cold drinks produced the
opposite response, with a reduction in sweat cancelling out the
cooling power of the drink.
So, sweat cancelling might be greater than the cooling effect of the drink.
However, this happens only if sweat evaporated completely:
The caveat is that your sweat must fully evaporate in order to produce
the desired cooling effect. If you're exercising hard, or wearing too
many clothes, or in a very humid environment, you may produce sweat
more quickly than it can evaporate, in which case it's no longer
desirable to ramp up your sweat rate further.
This tackles the thermodynamic aspect of drinking cold water. However, efficiency might measured through hydration. This article suggests that drinking cold water is more efficient than drinking warm water:
(..) cold water is absorbed more quickly into your body than warm
water, helping you rehydrate more quickly
As a conclusion, taken into the account both thermodynamics and hydration, I would say drinking cold water is more effective. Of course, if you do not have some throat issues that might get worse when drinking cold liquids.