Let $A$ be an abelian variety over a field $k$ of dimension $g$, and $H$ be a Weil cohomology theory for smooth projective varieties over $k$ with characteristic $0$ coefficient field $E$. 


Is it true that $\operatorname{dim} H^1(A)=2g$? I think this will follow from some standard conjectures, but do we know this unconditionally at present (at least for some low dimensional cases)? 


For example, one can prove $\operatorname{dim} H^1(A) \leq 2g$ by some power tricks, see Chapter 3. Theorem 8.1 in <https://pages.uoregon.edu/ddugger/wbook.pdf> (it may contain some typos, but the idea works). 


And if $k$ is algebraically closed and $A$ is superspecial, then $End\_k(A)\otimes \mathbb Q \cong M\_g(D\_{p,\infty})$ is central simple algebra of dimension $4g^2$ over $\mathbb Q$, hence $End(A) \otimes E \hookrightarrow End(H^1(A))$, and $dim H^1(A)=2g$.



Is $\operatorname{dim} H^1$ of an abelian variety the same for any Weil cohomology?
Yes, it's the same for all Weil cohomology theories --- see 2A.8 of Kleiman, S. L. Algebraic cycles and the Weil conjectures. Dix exposés sur la cohomologie des schémas, 359--386, Adv. Stud. Pure Math., 3, North-Holland, Amsterdam, 1968.


