Short answer: It goes down.
The Law of Demand says that, all other things equal, if price goes up then demand will go down. The percentage that quantity demanded goes down for a 1% increase in price is the price elasticity of demand.
But in the real world, it's not quite so simple. The Law of Demand is like the Pirate Code: It's really more like what you'd call a "guideline". It's probably true for 90% of goods, maybe even 95%.
But what about the rest?
There are some goods for which price is basically irrelevant, at least within a wide range; these are perfectly inelastic. If the price of salt quadrupled, would you buy less salt? Probably not---you need salt to live, and salt isn't that expensive anyway.
There are also some goods for which an increase in price will result in an increase in demand.
This is weird and counter-intuitive, so it bears some further explanation.
The first way this can happen is called a Giffen good. The idea is basically that if a good is an inferior form of a vital necessity, there are some circumstances for which raising its price will have an income effect of reducing the total amount of stuff you can buy that is larger than the substitution effect of making that particular good more expensive, and as a result you buy more of the inferior good.
As a hypothetical example, suppose you have $20 and you need four bags of grapes for a party.
Green grapes are $2 per bag and red grapes are $8 per bag. You like red grapes better, but you can't buy all red grapes because you only have $20. So you buy 2 bags of red grapes ($16) and 2 bags of green grapes ($4).
But now suppose that the price of green grapes rises to $4 per bag. You still can't afford to buy all red grapes. But now you can't even afford to go half-and-half like before. Your best choice is now to buy 3 bags of green grapes ($12) and 1 bag of red grapes ($8). The price of green grapes went up, but you bought more green grapes---because you couldn't afford to buy anything else.
Giffen goods are pretty rare in real life. The more common reason why rising price can result in rising demand is a Veblen good, which is a good that people buy simply to show off how rich they are. These are all over the place, from clothes to houses to cars---and arguably even lattes.
Why do people buy Lamborghinis? Is it really because Lamborghinis are such great cars that they're worth paying the price of a house? Probably not. The reason people buy Lamborghinis is to show that they can. If Lamborghini decided to lower the price of their cars to something more sensible, they might actually lose business, because it would no longer be as impressive a status symbol to own a Lamborghini. Thus making the price go down might actually result in less demand.
Finally, there is a third reason why rising price might not result in reduced demand, and that is speculation. The most obvious examples are stocks and gold. If the price of Apple stock goes up, do you buy less Apple stock? Not necessarily. If you think this is part of a coming trend, you might buy more Apple stock, hoping that the price will continue to rise and you can make money that way. You could be wrong---speculation is risky---but there are people who make huge amounts of money betting on price swings like this.