I think that in general $vol(X)$ is the best possible upper bound for the multiplicity of the first eigenvalue.

For certain arithmetic compact Riemannian surfaces (associated to division algebras), the first non-trivial eigenvalue is assumed to be larger or equal $1/4$. Some better upper bounds are due to Michel and Venkatesh via the Jacquet-Langlands correspondence. http://math.stanford.edu/~akshay/research/MV.pdf

I think that in general $vol(X)$ is the best possible upper bound for the multiplicity of the first non-trivial eigenvalue. This can be proved rigorously for compact Riemann surfaces with the Selberg trace formula, but I would guess this holds more general via an analysis related to Weyl laws.

For certain arithmetic compact Riemannian surfaces (associated to division algebras), the first non-trivial eigenvalue is assumed to be larger or equal $1/4$. Some better upper bounds in this particular case are due to Michel and Venkatesh (the Jacquet-Langlands correspondence has to be applied). http://math.stanford.edu/~akshay/research/MV.pdf

The Langlands correspondence also suggests that the multiplicity can be arbitrary large.