  The ideal gas equation of state (EOS) with a constant adiabatic index is a
poor approximation for most relativistic astrophysical flows, although it is
commonly used in relativistic hydrodynamics. The paper develops high-order
accurate physical-constraints-preserving (PCP) central discontinuous Galerkin
(DG) methods for the one- and two-dimensional special relativistic hydrodynamic
(RHD) equations with a general EOS. It is built on the theoretical analysis of
the admissible states for the RHD and the PCP limiting procedure enforcing the
admissibility of central DG solutions. The convexity, scaling and orthogonal
invariance, and Lax-Friedrichs splitting property of the admissible state set
are first proved with the aid of its equivalent form, and then the high-order
central DG methods with the PCP limiting procedure and strong stability
preserving time discretization are proved to preserve the positivity of the
density, pressure, and specific internal energy, and the bound of the fluid
velocity, maintain the high-order accuracy, and be $L^1$-stable. The accuracy,
robustness, and effectiveness of the proposed methods are demonstrated by
several 1D and 2D numerical examples involving large Lorentz factor, strong
discontinuities, or low density or pressure etc.
