First of all, he did NOT double cross them, strictly speaking.
The deal was for him to help the trio break into Lestrange's vault - NOT to help them get back out.
Griphook DID get them all the way into the vault so he fulfilled the letter of his bargain 100%.
For that matter, he even did MORE than the bargain and helped them get the cup while in the vault:
It was Griphook who had seen it [sword with the cup on it] and Griphook who lunged, and in that instant Harry knew that the goblin had never expected
them to keep their word.
Now, you may wonder why he didn't go above and beyond, and helped them get out which was NOT part of the bargain. For that, see the next section.
There is no direct canon proof, but it can be easily inferred:
Bill Weasley very explicitly told Harry about Goblin ideas of ownership:
“Then I have to say this,” Bill went on. “If you have struck any
kind of bargain with Griphook, and most particularly if that bargain
involves treasure, you must be exceptionally careful. Goblin
notions of ownership, payment, and repayment are not the same
as human ones.
... However, there is a belief among some goblins, and
those at Gringotts are perhaps most prone to it, that wizards cannot
be trusted in matters of gold and treasure, that they have no
respect for goblin ownership.
The Sword was pretty much one of the main Goblin made artifacts.
“Wizarding arrogance again! That sword was Ragnuk the First’s, taken from him by Godric Gryffindor! It is a lost treasure, a masterpiece of goblinwork! It belongs with the goblins. The sword is the price of my hire, take it or leave it!”
Griphook probably had an inkling that Harry intended to keep the sword, at least temporarily. And even if not, he would have considered it to be a natural move for a wand-carrier, even a slightly more evolved one like Harry.
Harry/Ron pretty much gave himself away when negotiating:
“I want the sword. The sword of Godric Gryffindor.”
Harry’s spirits plummeted.
“You can’t have that,” he said. “I’m sorry.”
“Then,” said the goblin softly, “we have a problem.”
“We can give you something else,” said Ron eagerly. “I’ll bet the Lestranges have got loads of stuff, you can take your pick once we get into the vault.”
He had said the wrong thing. Griphook flushed angrily.
“I am not a thief, boy! I am not trying to procure treasures to which I have no right!”
“The sword’s ours—”
And later - after taking an obvious half-time break for discussion:
Hermione frowned at the floor while he was speaking; he felt irritated at her, afraid that she might give the game away.
However, Griphook had eyes for nobody but Harry.
“I have your word, Harry Potter, that you will give me the sword of Gryffindor if I help you?”
“Yes,” said Harry.
“Then shake,” said the goblin, holding out his hand.
Harry took it and shook. He wondered whether those black eyes
saw any misgivings in his own. Then Griphook relinquished him,
clapped his hands together, and said, “So. We begin!”
... and later
The door of the bedroom opened and Griphook entered. Harry
reached instinctively for the hilt of the sword and drew it close to
him, but regretted his action at once. He could tell that the goblin
Therefore, as far as Griphook was concerned, this pre-emptive move was 100% logical and sensible.