Protostars grow from the first formation of a small seed and subsequent accretion of material. Recent theoretical work has shown that the pre-main-sequence (PMS) evolution of stars is much more complex than previously envisioned. Instead of the traditional steady, one-dimensional solution, accretion may be episodic and not necessarily symmetrical, thereby affecting the energy deposited inside the star and its interior structure. Given this new framework, we want to understand what controls the evolution of accreting stars. We use the MESA stellar evolution code with various sets of conditions. In particular, we account for the (unknown) efficiency of accretion in burying gravitational energy into the protostar through a parameter, ksi, and we vary the amount of deuterium present. We confirm the findings of previous works that, in terms of evolutionary tracks on an Hertzprung-Russell (H-R) diagram, the evolution changes significantly with the amount of energy that is lost during accretion. We find that deuterium burning also regulates the PMS evolution. In the low-entropy accretion scenario, the evolutionary tracks in the H-R diagram are significantly different from the classical tracks and are sensitive to the deuterium content. A comparison of theoretical evolutionary tracks and observations allows us to exclude some cold accretion models (ksi~0) with low deuterium abundances.