〜(よ)うが・〜(よ)うと
no matter (what / how), even if
Verb volitional + が / と (often with 何, どんなに, 誰が)
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〜うが and 〜うと after the volitional form mean "no matter what happens, it won't change things": 誰が何と言おうと、私の気持ちは変わらない, "whatever anyone says, my feelings won't change".
The first half often has a question word or どんなに, and the second half is a firm resolve or an unchanging fact.
It's a stronger, more defiant version of ても. The shape is the volitional: 言う → 言おう + と, 降る → 降ろう + と, される → されよう + が. い-adjectives use かろう: 高かろうが.
The two forms are interchangeable. Pair it with まい for "whether or not": 行こうが行くまいが (the next point).
Examples
誰が何と言おうと、私の気持ちは変わらない。
Whatever anyone says, my feelings won't change.
どんなに反対されようが、やめるつもりはない。
However much they oppose me, I've no intention of quitting.
雨が降ろうと、試合は行われる。
The match will go ahead even if it rains.
何が起ころうと、私は君の味方だ。
Whatever happens, I'm on your side.
周りにどう思われようが、気にしない。
I don't care what people think of me.