Density functional alloy theory is used to accurately describe the three core effects controlling the thermodynamics of random Cu-Au alloys. These three core effects are exchange correlation (XC), local lattice relaxations (LLRs), and short-range order (SRO). Within the real-space grid-based projector augmented-wave (GPAW) method based on density functional theory (DFT), we adopt the quasinonuniform XC approximation (QNA), and take into account the LLR and the SRO effects. Our approach allows us to study the importance of all three core effects in a unified way within one DFT code. The results demonstrate the importance of the LLR term and show that going from the classical gradient level approximations to QNA leads to accurate formation energies at various degrees of ordering. The order-disorder transition temperatures for the 25%, 50%, and 75% alloys reach quantitative agreement with the experimental values only when also the SRO effects are considered. © 2019 American Physical Society.