In John 19 it states: 

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> 6 *As soon as the chief priests and their officials saw him, they shouted, “Crucify! Crucify!”* But Pilate answered, “You take him and crucify him. As for me, I find no basis for a charge against him.” 7 The Jewish leaders insisted, “We have a law, and according to that law he must die, **because he claimed to be the Son of God.”** **8 When Pilate heard this, he was even more afraid, 9 and he went back inside the palace. “Where do you come from?” he asked Jesus, but Jesus gave him no answer. 10 “Do you refuse to speak to me?”** Pilate said. “Don’t you realize I have power either to free you or to crucify you?” 11 Jesus answered, “You would have no power over me if it were not given to you from above. Therefore the one who handed me over to you is guilty of a greater sin.” 12 From then on, Pilate tried to set Jesus free, but *the Jewish leaders kept shouting, “If you let this man go, you are no friend of Caesar. Anyone who claims to be a king opposes Caesar.”*
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Several commentators I have referred to seem to suggest, that Pilate's increased fear was based on his pagan religious beliefs in the potential that Jesus was some Herculean-like "son of a god." For example:

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> We need not suppose that Pilate was suddenly affected by the truth of
>  Jewish monotheism; but he may readily have believed that the wondrous
>  Being before him was enshrouded in a mystery of supernatural portent
>  and pretension that he could not fathom, and before which he trembled.
>  The idea of Divine energy enshrined in and wielded by human beings was
>  not altogether foreign to heathen thought . . . (Pulpit Commentary)
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and 

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> He before feared to shed innocent blood, and now he became more afraid
>  than ever to take his life; suspecting, probably, that the account
>  which he heard of him might be true, and that he might be a divine
>  person. For doubtless he had heard of some of the many miracles which
>  Jesus had performed, and now, it seems, began to think that perhaps
>  what had been currently reported was true, and that he really had
>  performed the wonderful works ascribed to him. For it is very well
>  known, that the religion which the governor professed directed him to
>  acknowledge the existence of demi-gods and heroes, or men descended
>  from the gods. Nay, the heathen believed that their gods themselves
>  sometimes appeared on earth, in the form of men . . . (Benson).
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It seems that Pilate up to this point is not very concerned with what is an internal Jewish matter, but with the new accusation (as per John's account) he (Pilate) seems more fearful. [I have emphasized in bold the section that the section that seems to support such a view].

However, there is what seems in John's narrative to be a growing agitation on the part of the Jewish leadership [in my italics] which raises the question: Could this growing fear be of a riot or tumult, rather than a superstitious response? [see Matthew 27:23-24]:

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> . . .But the mob roared even louder, “Crucify him!” 24 Pilate saw that he wasn’t getting anywhere and that a riot was developing.
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Based on cultural and historical understanding of the times, which explanation is the more probable: the fear was religious, or the fear was of unrest?