  The superconducting properties of the recently discovered double Fe$_2$As$_2$
layered high-$T_c$ superconductor RbCa$_2$Fe$_4$As$_4$F$_2$ with $T_c\approx$
30~K have been investigated using magnetization, heat capacity,
transverse-field (TF) and zero-field (ZF) muon-spin rotation/relaxation
($\mu$SR) measurements. Our low field magnetization measurements and heat
capacity (C$_p$) reveal an onset of bulk superconductivity with $T_{\bf c}\sim$
30.0(4) K. Furthermore, the heat capacity exhibits a jump at $T_{\bf c}$ of
$\Delta$C$_p$/$T_{\bf c}$=94.6 (mJ/mole-K$^2$) and no clear effect of applied
magnetic fields was observed on C$_p$(T) up to 9 T between 2 K and 5 K. Our
analysis of the TF-$\mu$SR results shows that the temperature dependence of the
magnetic penetration depth is better described by a two-gap model, either
isotropic $s$+$s$-wave or $s$+$d$-wave than a single gap isotropic $s$-wave or
$d$-wave model for the superconducting gap. The presence of two superconducting
gaps in RbCa$_2$Fe$_4$As$_4$F$_2$ suggests a multiband nature of the
superconductivity, which is consistent with the multigap superconductivity
observed in other Fe-based superconductors, including ACa$_2$Fe$_4$As$_4$F$_2$
(A=K and Cs). Furthermore, from our TF-$\mu$SR study we have estimated an
in-plane penetration depth $\lambda_{\mathrm{ab}}$$(0)$ =231.5(3) nm,
superconducting carrier density $n_s = 7.45 \times 10^{26}~ $m$^{-3}$, and
carrier's effective-mass $m^*$ = 2.45\textit{m}$_{e}$. Our ZF $\mu$SR
measurements do not reveal a clear sign of time reversal symmetry breaking at
$T_{\bf c}$, but the temperature dependent relaxation between 150 K and 1.2 K
might indicate the presence of spin-fluctuations. The results of our present
study have been compared with those reported for other Fe pnictide
superconductors.
