  We present a study of the exclusive $pp \to pp K^{+} K^{-} K^{+} K^{-}$
reaction at high energies. We consider diffractive mechanisms with the
intermediate $\phi \phi$ state with its decay into the $K^{+} K^{-} K^{+}
K^{-}$ system. We include the $\phi(1020)$ $\hat{t}/\hat{u}$-channel exchanges
and the $f_{2}(2340)$ $s$-channel exchange mechanism. This $f_{2}$ state is a
candidate for a tensor glueball. We discuss the possibility to use the $pp \to
pp \phi \phi$ process in identifying the odderon exchange. An upper limit for
the $\mathbb{P} \mathbb{O} \phi$ coupling is extracted from the WA102
experimental data. The amplitudes for the processes are formulated within the
tensor-pomeron and vector-odderon approach. We adjust parameters of our model
to the WA102 data and present several predictions for the ALICE, ATLAS, CMS and
LHCb experiments. Integrated cross sections of order of a few nb are obtained
including the experimental cuts relevant for the LHC experiments. The
distributions in the four-kaon invariant mass, rapidity distance between the
two $\phi$ mesons, special ``glueball filter variable'', proton-proton relative
azimuthal angle are presented. The distribution in rapidity difference of both
$\phi$-mesons could shed light on the $f_{2}(2340) \to \phi \phi$ coupling, not
known at present. We discuss the possible role of the $f_{0}(2100)$,
$\eta(2225)$, and $X(2500)$ resonances observed in the $\phi \phi$ channel in
radiative decays of $J/\psi$. Using typical kinematic cuts for LHC experiments
we find from our model that the odderon-exchange contribution should be
distinguishable from other contributions for large rapidity distance between
the $\phi$ mesons and in the region of large four-kaon invariant masses. At
least, it should be possible to derive an upper limit on the odderon
contribution in this reaction.
