The fuel-driven process of replication in living systems generates distributions of copied entities with varying degrees of copying accuracy. Here we introduce a thermodynamically consistent ensemble for investigating universal population features of template copying systems. In the context of copolymer copying, coarse-graining over molecular details, we establish a phase diagram of copying accuracy. We discover sharp non-equilibrium transitions between populations of random and accurate copies. Maintaining a population of accurate copies requires a minimum energy expenditure that depends on the configurational entropy of copolymer sequences.