What can be said about the space of vector fields for which a given, say $C^1$, function is a Lyapunov function?

Question

I am learning Morse homology and I have been thinking about the following observation. One way of doing, say finite-dimensional, Morse theory is by fixing a Morse function $f\in C^{\infty}(M)$, where $(M,g)$ is our given (orientable) Riemannian manifold, and then perhaps perturb the metric $g$ such that $(f,g)$ is a Morse Smale pair. With this data one can construct the homology $H_*(M)$ of $M$. Another approach is by fixing our Morse function $f$ and then using pseudo gradients to construct our flow, and eventually our homology groups. I never thought there was any substantial difference between the two approaches but it seems to me that the second approach is superior when one wants to consider higher algebraic structures on $H_*(M)$, simply because there should be a larger space of perturbations. In particular the pseudo gradient approach allows one to perturb the (un)stable manifolds in a way which changes the position of its tangent space at the critical point, which cannot be done by simply perturbing a metric in a gradient flow. However, understanding the flexibility of this approach requires one to understand the space of vector fields for which $f$ is a Lyapunov function. Does anyone know a reference or have any nice inputs?