[pdonis]

> is this similar on how integrals always have a constant as a free parameter that need to be determined by other means?

Not really, no. It's a consequence of the assumptions made when the Einstein Field Equation (EFE) is derived from a Lagrangian using the principle of least action.

The assumptions are that the Lagrangian should be a Lorentz scalar (which is required of any Lagrangian) and that it should include no more than second derivatives of the metric. The Ricci scalar R meets this requirement and is the Lagrangian that was originally used by David Hilbert to derive the EFE (without a cosmological constant). But a simple constant (the cosmological constant) also meets the requirement, and therefore should be included in the Lagrangian; including it leads to the cosmological constant term in the EFE.

[gpderetta]

Thanks for the explanation in this and the sibling comment!