The first question is what you need this checksum for. If you don't need the cryptographic properties, then a non-cryptographic hash, or a hash that is less cryptographically secure (MD5 being "broken" doesn't prevent it being a good hash, nor still strong enough for some uses) is likely to be more performant. You could make your own hash by reading a subset of the data (I'd advise making this subset work in 4096byte chunks of the underlying file, as that would match the buffer size used by SHA1Managed as well as allowing for a faster chunk read than you would if you did say every X bytes for some value of X).
Edit: An upvote reminding me of this answer, has also reminded me that I since wrote SpookilySharp which provides high-performance 32-, 64- and 128-bit hashes that are not cryptographic, but good for providing checksums against errors, storage, etc. (This in turn has reminded me that I should update it to support .NET Core).
Of course, if you want the SHA-1 of the file to interoperate with something else, you are stuck.
I would experiment with different buffer sizes, as increasing the size of the filestream's buffer can increase speed at the cost of extra memory. I would advise a whole multiple of 4096 (4096 is the default, incidentally) as SHA1Managed will ask for 4096 chunks at a time, and this way there'll be no case where either FileStream returns less than the most asked for (allowed but sometimes suboptimal) or does more than one copy at a time.