Jews and Christians often perceive everything that happens in the universe as God’s will. Hence, even if Pharaoh freely chose to harden his own heart it will still be considered God’s will. Because everything that happens in this universe, even if we choose freely, does not escape his will.
Further, if we observe Exodus 7:13 it tells us that Pharaoh had the free-will to cause his heart to grow hard.
You may also want to read the text from Jewish lenses. Pharaoh, the Egyptian God, has treated the Israelite’s ruthlessly, slaughtered their children and has held them back against their will. They have now reached a point where they believe that only their God - Yahweh - can change Pharaoh’s disposition because he is more powerful.
That explains why Pharaoh seems to be competing with Yahweh by attempting to replicate the signs and wonders that occurred. This, in turn, causes Pharaoh’s heart to be hardened.
An analogy by Norman Geisler and Thomas Howe might help here,
“the sense in which God hardened his heart is similar to the way the sun hardens the clay and also melts the wax. If Pharaoh had been receptive to God’s warnings, his heart would not have been hardened by God. But when God gave Pharaoh a reprieve from the plagues, he took advantage of the situation. “But when Pharaoh saw that there was a relief, he hardened his heart and did not heed them [Moses and Aaron], as the Lord had said” (Exodus 8:15).”
I hope this helps.