  This paper proposes a new class of mass or energy conservative numerical
schemes for the generalized Benjamin-Ono (BO) equation on the whole real line
with arbitrarily high-order accuracy in time. The spatial discretization is
achieved by the pseudo-spectral method with the rational basis functions, which
can be implemented by the Fast Fourier transform (FFT) with the computational
cost $\mathcal{O}( N\log(N))$. By reformulating the spatial discretized system
into the different equivalent forms, either the spatial semi-discretized mass
or energy can be preserved exactly under the continuous time flow. Combined
with the symplectic Runge-Kutta, with or without the scalar auxiliary variable
reformulation, the fully discrete energy or mass conservative scheme can be
constructed with arbitrarily high-order temporal accuracy, respectively. Our
numerical results show the conservation of the proposed schemes, and also the
superior accuracy and stability to the non-conservative (Leap-frog) scheme.
