Temperature has a big influence on the speed at which yeasts and bacteria ferment. But they don’t need to be kept warm all the time. Here are some pointers:
You should try to keep your starter warm during the four-day initial build-up stage because it gets going better when it’s not too cool. It will work at cooler temperatures only more slowly. Sometimes novice bakers can mistake this slowness for inactivity – or they just get bored waiting for something to happen! A cheap plant propagator from the garden centre can make a good sourdough incubator. But a word of warning: put a thick mat (wood or cork or similar) on the heated surface of the propagator and then put your starter on that. Direct contact with the heat mat may ‘cook’ (and therefore kill) some or all of your starter.
After your sourdough is established and you have used it, i.e. ‘refreshed’ it and then made a final dough with it, you can safely keep it in the fridge in a sealed container (not necessarily hermetically sealed but well closed so that cross-contamination from other fridge items cannot happen). It stays there until you next want to make bread, whereupon you refresh it again to re-activate the yeasts and bacteria.
You don’t need to ‘feed’ a starter. Those who talk about ‘feeding’ leavens and sourdoughs misconstrue the biology of yeasts and bacteria. These organisms are not like pets or children in requiring three meals a day. They don’t die once their immediate energy source is exhausted, but tend to go into a dormant state.
That said, prolonged storage of a leaven will result in it going more acid and it may benefit from an ‘intermediate’ refreshment to dilute the acidity and re-balance the ratio of yeasts and lactic acid bacteria. If you do an intermediate refreshment, remember to do the maths beforehand so that you don’t end up with too much refreshed leaven when you come to make the final dough.