  In this work, we systematically investigate the inflationary complexity of
the two-mode squeezed state with thermal effect for the single field inflation,
modified dispersion relation, and non-trivial sound speed with the method of
closed system and open system, respectively. Since the various quantum
gravitational framework could lead to this kind of modified dispersion relation
and non-trivial sound speed, so that our analysis is valid for most
inflationary models. $(a)$. The numeric of Krylov complexity in the method of
the closed system indicates that the evolution of Krylov complexity highly
depends on the squeezed angle parameter once taking the thermal effect into
account, which will decay into some very tiny values, but the Krylov complexity
will always enhance without thermal effect. $(b)$. The numeric of circuit
complexity shows that the evolution is always increasing no matter whether
there are thermal effects or not which is independent of the evolution of
squeezed angle parameter. $(c)$. By utilizing the method of open system, we
first construct the wave function. Our investigations show the evolution of
Krylov complexity will enhance upon some peaks factoring in the thermal effects
and the Krylov complexity will always increase without thermal effect. $(d)$.
We also calculate the Krylov entropy in the method of closed system and open
system, which indicates that the hotter the universe is, the more chaotic the
universe becomes. Furthermore, our derivation for the Krylov complexity and
Krylov entropy could nicely recover into the case of closed system under the
weak dissipative approximation, which confirms the validity of construction for
the wave function. Finally, our numeric of Lanczos coefficient shows that the
non-trivial sound speed has minimal chaos compared to the other two cases.
