  We study time-reversal-invariant topological superconductivity in topological
insulator (TI) thin films including both intra- and inter-surface pairing. We
find a nontrivial topology for multiple different configurations. For
intra-surface pairing a $\pi$-phase difference between the intra-surface
pairing states is required. We show that in this case the resulting topological
phase is highly tunable by both an applied electric field and varied chemical
potential. For spin-singlet inter-surface pairing, a sign-changing tunnel
coupling present in many TI thin films is needed, and again, the topology can
be tuned by electric field or doping. Notably, we find that the required
inter-surface pairing strength for achieving nontrivial topology can still be
subdominant compared to the intra-surface pairing. Finally, for spin-triplet
inter-surface pairing we prove that the superconducting state is always
topological nontrivial. We show that thin films of Cu-doped Bi$_2$Se$_3$ will
likely host such spin-triplet inter-surface pairing. Taken together, these
results show that time-reversal-invariant topological superconductivity is
common in superconducting TI thin films and that the topological phase and its
Kramers pair of Majorana edge modes is highly tunable with an applied electric
field and varied chemical potential.
