Pressure changes the RI of water. For details check the IAPWS formulation. Using two step procedure one can calculate thermodynamic properties of water for defined temperature and pressure, which can be used for calculation of RI. Diluted aqueous solutions should exhibit similar behavior.
Of course, RI of an aqueous solution can depend on the pressure. There are two possible channels of this dependence: (i) the refractive index of the water itself depends on the pressure (see http://www.nist.gov/data/PDFfiles/jpcrd282.pdf), (ii) the refractive index of the soluble substance also depends on the pressure varying from one substance to another.
RI of water definitely changes with pressure. There are several contributing factors to that change:
1. Changes in pressure: delta_n / delta_P = 0.16x10-4 (RIU/atm).
2. With increased pressure there is also change in temperature. Typical relationship is delta_n / delta_T = -10-4 (RIU/C).
3. Depending on your ambient environment changes in pressure lead to changes in gas solubility which in turn also affects RI.
4. Depending on the dimensions of your liquid reservoir (bathtub vs 5um capillary) bulk water has slightly different properties than interfacial water (near the surface).
5. I would suspect that in the absence of the flow, increased pressure would not affect the slope of your thermal gradient but rather have a DC offset.