At one time, even numbered months had 30 days and odd numbered months had 31 days. This adds up to 366 days. Julius Caesar thought it would be a nice gesture if the Roman senate named the fifth month after him. As they were not used to saying no to him, this was not a problem. However, when Augustus succeeded Caesar to the throne, he demanded that the sixth month be given his name. The senate swallowed hard, but agreed to do so. The problem was that the sixth month was an even month having 30 days. Augustus could not tolerate the fact that his month would have fewer days than Caesar's. The solution was to take a day from the twelveth month, which at that time was February, and add it to what we now call August. As a result, February had 29 days.
The Romans did the best they could to account for the fact that there were about 365.25 days and not 366 days in a year, but eventually their calendar got out of whack. The calendar we now use, the Gregorian Calendar named after Pope Gregory XIII, allows for February to have only 28 days, except an extra day is granted to February every fourth year. In addition, to keep things running as smoothly as possible, it was decided that even century years would not be leap years unless they were divisible by 400. The year 2000 is thus a leap year.
I've gotten way off the track. Since temperatures at the end of February are on average a little warmer than at the beginning of February, shouldn't leap year Februarys be a fraction of a degree warmer than non leap year Februarys? Looking at the temperature record for Washington D.C. from 1900 through the present, leap year temperatures for February are actually cooler than non leap year Februarys. This may not be the case for other cities in the U.S. If the Earth had no atmosphere and its axis were not tilted, the temperatures would only be a function of how close we are to the Sun. In this kind of environment, temperatures would be slightly warmer for those Februarys having 29 days.
The point is that the Earth's weather and climate are variable. For any one location it's not unusual for the temperature, from one week to the next, or one day to the next, to be quite different. If temperatures for a city were available for perhaps 1000 years, then there might be a very small warming noticable for leap year Februarys versus non leap year Februarys.
Like temperature, precipitation is also quite variable. You might ask "how can a month or a year that has more days, have less rain?" Of course, if it rained the same amount every day, then leap years would have more precipitation than non leap years, and if a long enough record were available, this would be observed just about everywhere. However, because of the variable nature of our weather, it's not at all unusual for any given leap year to have less rain than a given non leap year.
Come to think of it, why can't we move leap day from February to June. In that way, every fourth year, we would all have one more day of summer to enjoy and you would have one less day of school. I wonder if we can convince our friends in the Southern Hemisphere to go along with this.