[noun] + こそあれ [verb (dictionary form)] + こと + こそあれ
こそあれ grants that one thing is there and flatly denies its opposite: "if anything X, and certainly no Y". It follows a noun, and the clause after it is negative. A verb takes こと first.
あれ is the classical form of ある that こそ required. It survives in this pattern and never changes shape, whatever the tense of the sentence.
Sometimes こそあれ only concedes – "there may be X, but …" – and the second half need not be negative. The set phrase is ていど程度のさ差こそあれ, "to a greater or lesser degree".
こそあれ is written Japanese: essays, editorials, formal speeches. In conversation, a sentence with ても says the same thing: かんしゃ感謝はしても、うらんでは恨んではいない.