  We study the thermodynamic topology of 4D Euler-Heisenberg-AdS (EHAdS) black
hole and higher-order QED corrected Euler-Heisenberg-AdS black hole in
different ensembles using generalized off-shell free energy. In this approach,
black holes are viewed as defects in the thermodynamic space. We work in two
ensembles: canonical ensemble in which the charge is kept fixed and grand
canonical ensemble in which the conjugate potential $\phi_e$ is kept fixed. In
each case, the local and global topology of the thermodynamic space is
investigated via the computation of winding numbers at the defects. For 4D
Euler-Heisenberg-AdS black hole in canonical ensemble, the topological class is
found to be different depending on the Euler-Heisenberg (EH) parameter $a$. The
topological numbers for $a<0$ and $a>0$ cases are found to be $W=+1$ and $W=0$
respectively. The topological number is found to be independent of the
variation in pressure $P$ and charge $Q$ of the black hole. With the
introduction of higher order QED correction, the difference in the topological
class of 4D EHAdS black hole with the sign of $a$ is observed to go away.The
topological number in this case is found to be $W=+1$ irrespective of the
values of $a$, $P$ and $Q$. In the grand canonical ensemble, the topological
number for both EHAdS and higher order QED corrected EHAdS black hole is found
to be $W=0$, independent of the values of $P$, $\phi_e$ and $a$. Therefore, we
infer that the topological class of both 4D EHAdS black hole and higher order
QED corrected EHAdS black hole is ensemble dependent. Moreover, in the
canonical ensemble, higher order QED correction alters the topological class of
the black hole for positive values of EH parameter $a$. In the grand canonical
ensemble, the higher order corrections do not change the thermodynamic topology
of the black hole.
