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I have seen powerful spells that require a material component that costs a fair number of gold pieces. If a character chooses one of these spells, do they know what the cost is?

Lovell
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Boris
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Yes.

The player has knowledge of the item's cost, because they have read it in the rules.

The character has knowledge of the item's cost, in the same way they know how much a sword is worth, or any other object for that matter.

Whether or not the actual value of the item in the game lists its gold cost is up to you, and if you rule it's a more obscure material, perhaps the character has no knowledge of it. We're playing characters that exist within their own world, and they possess much knowledge we don't about the world they live in.

This is often reflected by things such as Nature, Religion, or History checks - a way to gauge whether or not your character is aware of something. On the other hand, the material components required to cast a spell is part of the valuable information they need to cast a spell; they need to know that the crystal ball for scrying, for example, has a value of 1,000gp or the spell will fail.

Robert Pain VanZant
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Sort Of

A wizard casting a spell which (in the rules) states some component with a minimum GP requirement would certainly know whether or not a specific component will work for that spell. Whether we choose the in-game reason to be because "it has sufficient magical resonance" or some other "wizard did it" hand-waving, or because in-game the GP requirement is the actual requirement is up to the DM. At our table we don't use the gold piece cost as the in-game reason, to guard against the following scenario:

Wizard's player: I need to cast this spell which requires a 100GP diamond, but I only have a 50gp diamond. Hey Barbarian, take this diamond and sell it back to me for 100GP.

Wizard: Casts spell with now-100gp-diamond

Gus
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    The rules already forbid the scenario you describe, so I'm not sure why you need a house rule to prevent it. – Thomas Markov Apr 30 '21 at 16:12
  • @ThomasMarkov can you point me to the rules that forbid it? I view that as a hole in the existing rule set, but it's entirely possible I've missed it/them. – Gus Apr 30 '21 at 16:44
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    It isn't as explicit as it should be, but costly material components always reference the "worth" of the item, which is how much it would go for at the market. The equipment section states of gems in particular: "These items retain their full value in the marketplace, and you can either trade them in for coin or use them as currency for other transactions." The mechanical value of a gem is static, and its suitability as a spell component is tied to that value, and has nothing to do with "how much you paid for it". The rules use this value as an abstraction of size and quality. – Thomas Markov Apr 30 '21 at 16:48
  • Suppose our friend said "I need a car worth at least $3400 for a project", If I sold you a 2008 Ford Focus that Kelly Blue Book lists as having a value of $1700, for $3400, would our friend say "this car is worth $3400"? Of course not. It's worth $1700, and our friend would be very disappointed in you. – Thomas Markov Apr 30 '21 at 16:55
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    For a more thorough explanation, see my answer here: How big is a 5000 gp diamond? and also What makes a spellcasting component worth x gp? – Thomas Markov Apr 30 '21 at 16:58
  • Interesting commentary, that has revealed yet another habit-from-older-rules-versions that my group unknowingly carried forward. Our found gems never sell for book value -- functionally their worth is whatever we can convince someone to pay for it. I think I'd prefer the gem X = 25gp simplification, though it seems to make gems that can't be used in spells a bit pointless. – Gus May 03 '21 at 16:09