There are number of different kinds of “bean paste”, but since you mention the plastic carton, am I assuming correctly it’s “doenjang” (a fermented soybean paste)? That’s analogous to the Japanese product red miso, so between the salt content and it being full fermented, it will stay “good to eat” for a fairly long time in the fridge after it’s opened, and “safe to eat”, probably until/unless it gets moldy. Some people consider that this and miso last “forever”, but in my own opinion, the flavor starts to suffer after 2-3 months. (If you use it only rarely, you can freeze, and it will stay fresh-tasting much longer that way. And if you freeze it in small portions, you don’t have to worry about letting it thaw out enough to scoop what you need before putting it back in the freezer.)
At room temprature, it will keep for “a while”, but the flavor will decline more rapidly. I would not expect it to keep “good flavor” for more than a month or so at temperatures in the high 60s (Fahrenheit, or the mid-teens Celsius) or warmer, though as long as you don’t see any visible signs of mold or bacterial growth on the surface, it should be “safe” at room temperature as well.
