Graham Ellis would be able to better comment on the correctness of his code for $SL(5,\mathbb Z)$, as he appears to be the author of the HAP package in GAP.
But his code executes quickly and claims to compute the homology of $SL(5,\mathbb Z)$.
$$H_2(SL_5(\mathbb Z), \mathbb Z)$$
LoadPackage("HAP");;
R:=ResolutionArithmeticGroup("SL(5,Z)",4);;
Homology(TensorWithIntegers(R),2);
Returns $[2, 12]$, which I believe means the code is claiming this homology $H_2$ group is isomorphic to $\mathbb Z_2 \oplus \mathbb Z_{12}$. It similarly computes $H_3$ to be $\mathbb Z_2 \oplus \mathbb Z_{24}$, and $H_4$ to $\mathbb Z_{12}$.
I believe the 4 in the call for the resolution is a reference to the length of the resolution, so 4 is longer than required for $H_2$. With the resolution of length 4 call, the code chokes on an $H_5$ computation.
As was pointed out in the comments these answers disagree with what's known in the literature. After digging around in Graham Ellis's documentation a little more I found the comment regarding the ResolutionArithmeticGroup command:
Inputs a positive integer n
and a string P
equal to one of the following:
"SL(2,Z)" , "SL(3,Z)" , "PGL(3,Z[i])" , PGL(3,Eisenstein_Integers)" , "PSL(4,Z)" , "PSL(4,Z)_b" ,"PSL(4,Z)_c" , "PSL(4,Z)_d" , "Sp(4,Z)"
So it would appear the answer to the question of this thread is no, GAP currently can't compute group homology of $SL_5(\mathbb Z)$.
The documentation is here: https://docs.gap-system.org/pkg/hap/doc/chap0_mj.html
Click on Chapter 5 to find the above.
It's computing something, just not the homology of the group.