We investigate the invariance of the Gibbs measure for the fractional Schrodinger equation of exponential type (expNLS) $i\partial_t u + (-Δ)^{\fracα2} u = 2γβe^{β|u|^2}u$ on $d$-dimensional compact Riemannian manifolds $\mathcal{M}$, for a dispersion parameter $α>d$, some coupling constant $β>0$, and $γ\neq 0$. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case $γ>0$, the measure is well-defined in the whole regime $α>d$ and $β>0$ (Theorem 1.1 (i)), while in the focusing case $γ<0$ its partition function is always infinite for any $α>d$ and $β>0$, even with a mass cut-off of arbitrary small size (Theorem 1.1 (ii)). (ii) We then study the dynamics (expNLS) with random initial data of low regularity. We first use a compactness argument to prove weak invariance of the Gibbs measure in the whole regime $α>d$ and $0<β< β^\star_α$ for some natural parameter $0<β^\star_α\sim (α-d)$ (Theorem 1.3 (i)). In the large dispersion regime $α>2d$, we can improve this result by constructing a local deterministic flow for (expNLS) for any $β>0$. Using the Gibbs measure, we prove that solutions are almost surely global for $0<β\llβ^\star_α$, and that the Gibbs measure is invariant (Theorem 1.3 (ii)). (iii) Finally, in the particular case $d=1$ and $\mathcal{M}=\mathbb{T}$, we are able to exploit some probabilistic multilinear smoothing effects to build a probabilistic flow for (expNLS) for $1+\frac{\sqrt{2}}2<α\leq 2$, locally for arbitrary $β>0$ and globally for $0<β\ll β^\star_α$ (Theorem 1.5).