We know that there can be possible higher derivative corrections (stringy corrections) to the Einstein-Hilbert action. In GR, to ensure that we get the Einstein Field equations from varying the E-H action, we necessarily have to add the Gibbons-Hawking-York boundary term (Wikipedia link to GHY), as well as a counterterm (to ensure the action is finite).

Let's say we consider adding higher derivative corrections to the Einstein-Hilbert action, i.e we include the Gauss-Bonnet term, as well as the (Riemann)^3 term and so on and so forth. However, would we need further boundary terms, and perhaps counterterms too beyond the GHY term to ensure that we obtain the correct E.O.M from $\delta S=0$.

To summarize my question: What is the nature of these boundary terms (beyond the GHY term)?

We know that there can be possible higher derivative corrections (stringy corrections) to the Einstein-Hilbert action. In GR, to ensure that we get the Einstein Field equations from varying the E-H action, we necessarily have to add the Gibbons-Hawking-York boundary term (Wikipedia link to GHY), as well as a counterterm (to ensure the action is finite).

Let's say we consider adding higher derivative corrections (also known as stringy corrections) to the Einstein-Hilbert action, i.e we include the Gauss-Bonnet term, as well as the (Riemann)^3 term and so on and so forth. However, would we need further boundary terms, and perhaps counterterms too beyond the GHY term to ensure that we obtain the correct E.O.M from $\delta S=0$.

To summarize my question: What is the nature of these boundary terms (beyond the GHY term)?