I found this expression: to put on the thinking cap, What does it mean and how to use?
It means to give deep consideration to a problem, or to brainstorm, or to ponder:
Gentlemen, this is a serious issue. We need to put on our thinking caps and come up with a good solution.
The thinking cap is the cap that Gyro Gearloose wears when he needs to find a solution for a problem. To put the thinking cap means then to find the solution, think about, reflect on.
If you're going to use it in conversation the usual ways would be:
[That's a difficult problem.] I'll go put on my thinking cap.
... I'll go away and think about it.
[That's a difficult problem.] Why don't you put on your thinking cap and we'll talk later?
... Why don't you go away and think about it and we'll talk later?
The general idea it expresses is that you're taking time just to sit and think hard about something.
You can also say things like
I've got my thinking cap on.
I'll put my thinking cap on.
and so on.
It is a sligthly humorous phrase and, as such, would not be used in a strictly formal setting (e.g. a meeting with the Queen).