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 $T_capprox$ 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 carriers effective-mass $m^*$ = 2.45textit{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.
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