For a definition of and some values of API gravity, go here:
Now, note that specific gravity is a function of specific gravity which is a function of density.
The real question is how are you determining API gravity?
If you are measuring the density and calculating API then yes, density will change with pressure.
The calculations are defined in the Manual of Petroleum Measurement Standards Chapter 11 section 1.
The thing is, if you are measuring at 1000psig then the measurement device will also have a pressure correction and you need to be sure it is properly configured.
For example, vibrating tube density meters are affected by pressure because they actually respond to a change in mass. The resonant frequency is a function of the mass of the system.
The mass changes when the density changes. It also changes when the pressure changes not just because the volume of fluid contained will increase but also because it affects the stiffness of the tube which then affects the resonant frequency.
Default configurations may assume a low pressure application and have a fixed pressure setting (there is often facility for a live pressure input but it may not be configured, if the pressure doesn't vary then you can specify the working pressure but you still need to configure the calculations).
The calculations need to be configured also to apply a pressure correction to the measured density (as per the calculations mentioned).
The only thing to work out is if the pressures you are working at show a significant density change because at low pressures the difference is often not worth worrying about, dependent on the application.
JMW