$TD=3; ℏ /: (ℏ^(d_. ))/;d>$TD :=0;

q=(E^(ℏ α β));

(Subscript[ϕ, 0])[x_] := (E^x);
(Subscript[ϕ, k_])[x_] := Module[{t},Simplify[(((E^t)-Normal@Series[(E^t),{t,0,k-1}])/(t^k)) /. t->x]];

K=3; $TD=5;
GS1=μ T((A^(-1))(Subscript[ϕ, 1])[ℏ μ(q-1)x]-1)-μ(A (Subscript[ϕ, 1])[ℏ μ((q^(-1))-1)x]-1) /. {A->(E^(-ℏ β a))}

K=3;
GS2=Collect[Normal@Series[GS1, {β,0,K}],β, Factor]

Series[GS1-GS2, {ℏ,0,$TD}]

Series[Normal@Series[GS1-GS2, {ℏ,0,$TD}], {β, 0,K}]

Normal@Series[GS1-GS2, {ℏ,0,$TD}]+((O[β])^(K+1))

Series[GS1-GS2, {β,0,K}]

Clear[ℏ]