Take a bucket of hot water and the other bucket of cold water. Why does the bucket full of cold water weigh more than bucket full of hot water?
Heating almost any material will cause it to expand. That is, its density will go down, as the same mass of material takes up more space. Or, alternately, the same volume of the material weighs less.
If you take your bucket and fill it to the brim with cool water then heat the water (not to boiling), some of it will spill out of the top.
PS: Water is one of a very few materials that does not follow this rule perfectly. Water is actually at its densest (and heaviest for a given size container) at 4C. If you compare a bucket of 4C water to a bucket of 1C water, the warmer bucket will be heavier.
Well, to set the stage, let's assume these are identical buckets, and they're both filled with the same pure H2O; get distilled water from the grocery store if you're actually trying this, as tap and most bottled water contain other chemicals that are helpful for drinking water, less so for science experiments. Let's also assume a couple more things: we've put the water in the buckets after getting the water to the desired temperatures, and that we seal the tops of the buckets as soon as we filled them.
So we have two identical sealed buckets, filled with water, the only difference being that we filled one with hot water and the other with cold water. As Sparr said already, heating most anything will cause it to expand; the molecules will space themselves out more, and that (with the exception of a few degrees before it freezes) holds true for water, too!
When we filled the two buckets, although it looked like we put the same amount (volume) of water in both of them, we actually fit more water molecules in the cold bucket than we did in the hot one because the molecules in hot water are more spaced out than they are in cold water. And since the mass of any water (H2O) molecule is the same (they all use the same basic building blocks), the bucket that we could fit more water molecules in—the cold bucket—is going to be the heavier one.
To further demonstrate this effect, you could let the hot bucket cool down to the temperature of the cold bucket. Unless something goes wrong, once the hot bucket has cooled to the temperature of the cold bucket, if you take off the lid, you should see that the bucket is no longer filled to the brim. Of course, the same number of molecules are still in the bucket—where else could they have gone if the bucket was sealed?—but as the water cooled, the molecules got closer together; the density increased. And the cold water you'd need to add to fill it back up to the brim would equal the weight of the cold bucket minus the weight of the hot bucket.
Assuming that the buckets are identical, the number of water molecules (and dissolved gasses) are identical. When placed upon a pair of scales, the conducted heat from the hot bucket into the surrounding air causes significant turbulent air motion upwards (hey, when have buckets been designed like aircraft wings) , the friction of the warm air moving upwards will place force on the bucket to move it the same direction - the scales will be tipped such that the hot bucket of water appears lighter.
It's important to be specific in your question: If you simply want to know about dissolved gasses, then in earth's atmosphere, the cold bucket will weigh more, as it will contain a greater proportion of dissolved especially CO2 and O2.
If you want to ask about the density of water at different temperatures, then apart from it's quirky behavior around -4 Celsius as ice, then you may as well compare a bucket of "water" at approaching 0 Kelvin to the similar bucket of "water" at 500 kelvin (Very hot Steam).
Back to room temperature: Assuming water, gasses, are both identical in the buckets. The increase in kinetic energy of the water molecules in the hot one and assuming E=mC^2 then the increased motion in the hot bucket would make it heavier albeit probably beyond the threshold of measurement at this time.
I would like further clarification of the question.