Air density vs CR

Will there be any extensive change in the compression ratio with the change in air density?

Some of this has to do with the semantics of how you define "compression ratio."

Usually people refer to a compression ratio set according to the piston design, head gasket, combustion chamber, etc. That's the geometric compression ratio. So you might have an engine that is nominally 10.0:1 compression. That compression ratio can be considered according to the valve timing however. If I close my intake valve at bottom dead center (540 degrees ATDC firing), then my effective compression ratio is basically 10.0:1, and that's how very old engines used to work (100 years ago) before they figured out valve timing tricks. If I have a long duration cam or variable valve timing setting that closes the intake valve at 80 degrees after bottom dead center (620 ATDC firing), it will be much lower effective compression. See attached file.

The total amount of air captured inside the cylinder could be referred to as the trapped air mass or air mass per charge or some other such term. That will increase with the density of the air. This will change your engine load calculation. For example, on a GM ECU, a lot of the maps are based on the "milligrams per charge" calculation for engine load. That will increase with more dense air.

The compression ratio is a mechanical aspect of the engine's design and this won't change with atmospheric conditions. What will change with air density is the air mass in the cylinder.