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I hope you appreciate that this is not such a simple operation to perform in general.

The simplest way is probably to replace $\dot{a}$ by $Ha$.

We can use the class I gave as an answer to Substituting a differential equation into an expression:

sage: eq1 = E1[0,0].expr().expand() == 0
sage: var('h')
sage: pde = diff(a.expr(), t) == h*a.expr()
sage: s = SubstituteEvolutionaryPDE(pde, t)
sage: s(eq1)
3*h^2 - Lambda + 3*k/a(t)^2 == 0

This also works when there are second or mixed derivatives, e.g.:

sage: eq2 = diff(a.expr(), t, t) + 13 == 0
sage: s(eq2)
h^2*a(t) + 13 == 0