  This work addresses some asymptotic behavior of solutions to the stochastic
convective Brinkman-Forchheimer (SCBF) equations perturbed by multiplicative
Gaussian noise in bounded domains. Using a weak convergence approach of
Budhiraja and Dupuis, we establish the Laplace principle for the strong
solution to the SCBF equations in a suitable Polish space. Then, the
Wentzell-Freidlin large deviation principle is derived using the well known
results of Varadhan and Bryc. The large deviations for short time are also
considered in this work. Furthermore, we study the exponential estimates on
certain exit times associated with the solution trajectory of the SCBF
equations. Using contraction principle, we study these exponential estimates of
exit times from the frame of reference of Freidlin-Wentzell type large
deviations principle. This work also improves several LDP results available in
the literature for the tamed Navier-Stokes equations as well as Navier-Stokes
equations with damping in bounded domains.
