Quick question

So I started curing my buds in 32 oz mason jars, the hygrometer is reading at 62%. My question is, while storing my jars somewhere will the temp and humidity outside the jar effect the cure?

Outside humidity will have no effect if the jar is sealed. Heat will degrade the THC, as will light. For long term storage, a cool, dark place is best. If you've got room in your freezer, even better!

make sure you burp them babies

I’ll echo the previous post about burping. The hygro can read 62% and U can still have problems. When u first seal up the buds, the moisture will begin to redistribute from the inside to the outside. This is called sweating….and it needs to happen.
Quick frequent burps are not the answer…..After the buds have been sealed for 25-36hours, open the jar and spread the buds out evenly on a screen or flat surface. The outside of the buds should feel a bit more moist.
Let them sit out an hour or 2….depending on the conditions of the room.
Allow them to dry a bit, then seal them back up. U may have to repeat this process a couple times before they are ready to store (and cure).
The sweating process redistributes the moisture more evenly and mitigates the chances of mold.