[Where] does brew store meta information?

[jsejcksn]

...for example, the timestamp of the last time that brew update was executed (or formulae were updated)

[jonchang]

This can be discovered by reading the source code. In this case, brew checks the mtime of the core tap directory.

[jsejcksn]

Thanks for responding @jonchang, although I'm not positive about what you're referencing. Is this what you mean?

https://github.com/Homebrew/brew/blob/e30ed42439f4cca705be54aa14193f6f0a05420d/Library/Homebrew/brew.sh#L540-L552