This is an old idea that has been investigated thoroughly by Patodi and Dodziuk. See their paper for more details. But what they proved is more subtle (about Hodge Laplacian vs metric Laplacian). Later Muller used their techniques to prove his part of the Cheeger-Muller theorem.

For the volume ratio analogy I think I answered in the previous post. The paper I mentioned there specifically showed that minus some exceptions, we should have (2.6.1) $$ \frac{\log(T_{an}M_{i})}{\textrm{volume}(M_{i})}\rightarrow -\frac{1}{6\pi} $$ if their conjecture holds based on another result of Muller. But I think some counter-examples have been found already.

This is an old idea that has been investigated thoroughly by Patodi and Dodziuk. See their paper for more details. But what they proved is more subtle (about Hodge Laplacian vs metric Laplacian). Later Muller used their techniques to prove his part of the Cheeger-Muller theorem.

For the volume ratio analogy I think I answered in the previous post. The paper I mentioned there specifically showed that minus some exceptions, we should have (2.6.1) $$ \frac{\log(T_{an}M_{i})}{\textrm{volume}(M_{i})}\rightarrow -\frac{1}{6\pi} $$ if their conjecture holds based on another result of Muller. But I think some (non-artihemetic) counter-examples have been found already.