Some kind of pump noise?

Sounds like the brake pressure pump. It's hydraulic not vacuum. You can verify this if you just press on the brake without starting the car. It will probably run.

 

I believe that is the pump that fills the thermos bottle with hot engine fluid. Mine does that as well, not to worry. Hal

 

Its the coolant transfere valve........Mine also cycles more than ever on my third year (at 37k miles) of ownership........It started around the second year of ownership..at approx 20k miles but was very rare............I would better describe the sound (as heard from the drivers seat) as hee.haw....hee.haw...hee.haw...hee.haw........from under the hood is sounds like click.clack...click.clack...click.clack.....................................I have no warning lights (and prius runs tip top getting 55mpg) and hee.haw...hee.haw... sound cannot be heard over the ac/heater fan if set at 3 or above (fan speed).....................................This valve is a three position valve attached to three hoses........Its purpose is to switch head coolant between heater core and thermos bottle.................It has some kind of resistance encoder that provides three different output voltages (2.5, 3.5, 4.5 volts) so that ecu knows which position the valve is in............It seems that the encoder goes bad in position 3 (prius fully warmed) normally routing heated head coolant to both heater core and thermos bottle simutaniously........................The ecu cannot determine the valves position due to intermittand encoder voltages valve the result is the ecu keeps switching valve from one position to another trying to compute the valve position..........................The documented functions of the valve are PREHEAT where heater core is blocked off allowing thermos coolant to only enter the ice................................The next function is ENGINE WARM-UP where the valve blocks flow to the thermos bottle and opens flow to the heater core..................................The next valve function is DRIVING where flow is both to heater core and thermos bottle..................... The next function is POWER-OFF where the thermos bottle route is closed the same as warm-up position.................The only real problem I've been able to find on PC regarding this valve is some ones heater stopped working the dealer replaced the valve said is wasn't a waranted item.....................I suspect the valves position encoder is something like a variable resistor (pot) whose wiper has scrapped all the resistive coating away form the encoder wheel (in the normal run position) due to driving vibrations, road bumps..etc...........................In my prius the coolant transfere works fine as indicated by scan guage monitoring water temp, engine temp...............So its my opinion that the valves encoder goes bad only in its final run position where is spends most of its life........But as the valve keeps cycling is sure to wear out other encoder position possibly resulting in no heater core heat.............. I'm sorry if this post is hard to read....I've have problems with prius chat, yahoo mail removing my carrage returns, tabs, etc. making the text all one big paragraph.....