Your equation f_OpenLB = f – f^eq means Cell[iPop] = f_iPop – t_iPop? or it is something else?
In openLB, I have found two expressions to calculate pressure.
1) p = rho / invCs2 which can be reduced to p = rho * Cs2
2) p = (rho-1) * Cs2.
Do you mean 1) is the usual pressure and 2) is the special pressure only for the codes of openLB?