Is it easier to simulate compressible flow than incompressible flow? If I want to simulate an incompressible flow, I need to solve pressure p by a Poisson equation (which is hard to solve numerically). But if it is a compressible flow, the pressure equation can be an EOS (equation of state) (which is simple). So is it in general easier to simulate a compressible flow?
 A: It depends. The incompressibility helps you to get rid of sound waves. Often they are just a nuisance and limit your time-step. Especially in low Mach number flows the sound waves are much faster than the flow and can be very limitting. E.g., in meteorology they must be filtered even when solving the compressible equations. 
The incompressible Navier-Stokes equations (or the related Boussinesq or anelastic approximation) are than used to simplify the process in some sense. I would argue that many numerical discretizations and boundary conditions are also simpler for incompressible flow.
Regarding the Poisson problem, the P. equation is actually pretty simple, but the associated linear system may be slow to solve iteratively. On regular grids you find in many atmospheric applications you can use FFT in the fast Poisson solvers.
A: I would say no, it is not easier.
The equation of state you mentioned will involve new variable(s) as the temperature for instance. Therefore you should add an equation for energy in the Navier-Stokes system to find the evolution of the temperature field.
Dealing with compressibility means you have density variations. You may also have to solve the continuity equation. Some common assumptions as anelastic behaviour or low Mach-number can help reducing the computational cost but it could lead to suppress the phenomenon you wanted to catch in the first place. 
As mentioned by Vladimir F, you can have acoustic waves or shock waves to take into account. It will also increase the simulation cost.
Simulate a compressible flow may not just be as simple as solving an EOS. I think you will unavoidably have a higher computational cost and maybe more numerical problems to deal with due to the more complex physical phenomenon.
(I cannot comment on Vladimir F answer due to low reputation but Boussinesq approximation is not identical to incompressibility hypothesis. It allows to keep a small variation of density around a reference value in a incompressible flow. It helps studying some buoyancy effects for instance. See Batchelor, Canuto, Chasnov on homogeneous buoyancy-generated turbulence for instance.)
