  Lensing of the Cosmic Microwave Background (CMB) changes the morphology of
pattern of temperature fluctuations, so topological descriptors such as
Minkowski Functionals can probe the gravity model responsible for the lensing.
We show how the recently introduced two-to-two and three-to-one kurt-spectra
(and their associated correlation functions), which depend on the power
spectrum of the lensing potential, can be used to probe modified gravity
theories such as $f({R})$ theories of gravity and quintessence models. We also
investigate models based on effective field theory, which include the
constant-$\Omega$ model, and low-energy Ho\vrava theories. Estimates of the
cumulative signal-to-noise for detection of lensing-induced morphology changes,
reaches ${\cal O}(10^3)$ for the future planned CMB polarization mission
COrE$^{+}$. Assuming foreground removal is possible to $\ell_{max}=3000$, we
show that many modified gravity theories can be rejected with a high level of
significance, making this technique comparable in power to galaxy weak lensing
or redshift surveys. These topological estimators are also useful in
distinguishing {\em lensing} from other scattering secondaries at the level of
the four-point function or trispectrum. Examples include the kinetic
Sunyaev-Zel'dovich (kSZ) effect which shares, with lensing, a lack of spectral
distortion. We also discuss the complication of foreground contamination from
unsubtracted point sources.
