I have problems to verify your calculations with Macaulay 2, concretely the fiber over $t = 0$ (resp. $a=0$ as its called by me).

I used the description given at the OP for the ideal $\mathcal{I}$ and computed it as id1 by elimination. With the map phi I set $a=0$ and got the ideal id11 in S2=QQ[w_0..w_4]. This is the homogeneous coordinate ring of the $\mathbb{P}^4_{QQ}$ which is the fiber of $\mathbb{P}^4_{QQ[a]}$ over $a=0$.

Calculating cohomology in $\mathbb{P}^4_{QQ}$ I get all cohomologies zero for id11.

I would be very happy if someone could reconcile my calculation with the results above and find a possible mistake that I have made.

A=QQ[a,Degrees=>{1:{}}];


S=A[x_0..x_4];

T=A[t,u];

phi = map(T,S, {t^4, t^3*u, a* t^2 * u^2, t * u^3, u^4});

id1 = ker phi;

                                        2                             2          3      2     2    2     3    2       2    2           2        2     2        2
ideal (x x  - x x , - x x  + a*x x , a*x  - x x , - x x  + a*x x , a*x  - x x , x  - x x , x x  - x x , x  - x x , - x  + a x x , - x x  + a*x x , - x x  + a*x x )
        1 3    0 4     2 3      1 4     3    2 4     1 2      0 3     1    0 2   3    1 4   0 3    1 4   1    0 3     2      0 4     2 3      0 4     1 2      0 4

S2=QQ[w_0..w_4]

phi=map(S2,S,{w_0,w_1,w_2,w_3,w_4})


id11 = phi(id1)

                                                 3      2     2    2     3    2      2      2    2
ideal (w w  - w w , -w w , -w w , -w w , -w w , w  - w w , w w  - w w , w  - w w , -w , -w w , -w w )
        1 3    0 4    2 3    2 4    1 2    0 2   3    1 4   0 3    1 4   1    0 3    2    2 3    1 2

i73 :  for i from 0 to 4 list prune HH^i(sheaf module id11)

o73 = {0, 0, 0, 0, 0}

o73 : List

i74 : for i from 0 to 4 list prune HH^i(sheaf S2^1/id11)

         1
o74 = {QQ , 0, 0, 0, 0}

o74 : List