is this really true that speed of light is same as we look from different parts of universe? According to the theory of relativity. The speed of light is same for every observer. But how can we know that this applies to more than a local region? What if light travelled to us from a distant region where the speed of light was different or where the relativistic principle did not apply?

Can we design an experiment that refutes both these ideas?
According to theory of relativity . Light speed is same for every observer . How can we prove it is right?


By doing experiments. Pela mentioned the Michelson-Morley experiment. That's what got Einstein thinking. That's why he came up with the theory of relativity. However other experiments will prove that the speed of light is not the same for every observer. It's the same for every inertial observer, because of the wave nature of matter. But see Is The Speed of Light Everywhere the Same? on the PhysicsFAQ website. It's written by Don Koks, a physicist in Adelaide. He says this:

"That the speed of light depends on position when measured by a non-inertial observer is a fact routinely used by laser gyroscopes that form the core of some inertial navigation systems.  These gyroscopes send light around a closed loop, and if the loop rotates, an observer riding on the loop will measure light to travel more slowly when it traverses the loop in one direction than when it traverses the loop in the opposite direction.  This is known as the Sagnac Effect.  The gyroscope does employ such an observer: it is the electronics that sits within the gyro.  This electronic observer detects the difference in those light speeds, and attributes that difference to the gyro's not being inertial: it is accelerating within some inertial frame.  That measurement of an acceleration allows the body's orientation to be calculated, which keeps it on track and in the right position as it flies".  

Your measurement of the speed of light is not totally constant, it depends on your acceleration. If you were in a room in an accelerating rocket, you would be what Koks called the "global" observer, and you would say that light at the ceiling travels faster than light down near the floor.


  But how can we know that this applies to more than a local region? 


By understanding general relativity. See the general relativity section of the PhysicsFAQ article: 

"Einstein talked about the speed of light changing in his new theory.  In the English translation of his 1920 book "Relativity: the special and general theory" he wrote: "according to the general theory of relativity, the law of the constancy of the velocity [—Either Einstein or his translator obviously mean "speed" here, since velocity (a vector) is not in keeping with the rest of his sentence. People often say "velocity" when they clearly mean "speed".] of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity [...] cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity [speed]  of propagation of light varies with position." This difference in speeds is precisely that referred to above by ceiling and floor observers." 


  What if light travelled to us from a distance region where the speed of light was different or where the relativistic principle did not apply? 


You wouldn't measure any difference, because you measure the speed of light here, not somewhere else.