England ace Hart, 23, has broken into the first team for club and country this term.
Cech said: "It's been difficult for him because he has had to show everyone that he deserves to play ahead of such a highly-rated keeper as Shay Given.
"He started with a fantastic game at Tottenham and then he really came into the spotlight when he started playing for the national team.
"So now, all of a sudden, everyone is looking at him in a different light.
"He's no longer just one of the candidates for an international call-up, he is the No 1 for England and for a club with huge ambitions.
"He has much more pressure on him now but has done really well so far."