BobK is alluding to two issues with switching.
The first is that while the mosfet switches, its resistance changes from Rds(on) (or something similar) to infinity (or something effectively the same). During this process, the current through the mosfet falls. HOWEVER, the power dissipation first rises, before falling again. This is because the power is related to the current squared across a resistance.
The upshot of this is that during the switching period, the mosfet dissipates more energy (possibly far more energy per unit time) than if it is on or off. So the more often you switch it on and off, the more time it spends switching, and the greater power is lost. SO... The mosfet gets hotter, and the load gets less power.
This is the first consideration. How much power is lost in the mosfet?
I'm assuming a 12V supply and a 1A load.
Well, if you're switching it on and off 20 times per second (10Hz), and each time you turn it off (10 times) it takes 10us, then it is switching for 100us out of each 1000000us, or 0.01% of the time.
Imagine that during switching, we calculate the losses to be about 3W. That's 3W per second, so over a second that's 0.3mW. For completeness, lets assume that while on, the Rds is 0.1 ohm and that the duty cycle is 50%. So the power lost while the device is on is 0.1*1^2*0.5, which is 0.05W (50mW)
This means the total power lost in heat is about 50.3mW. The loss due to switching is tiny, and as Bob suggests, can be ignored.
Now let's try that at 10kHz...
now there are 10,000 switching (off) events each taking 10uS. This means that the device is now switching 10% of the time. So the switching losses rise to 0.3W, and lets assume the other losses remain the same.
So at 10kHz, the dissipation rises to 50mw + 300mW = 350mW. The switching losses dominate. But, the device does get turned on and off -- it's just that your mosfet gets hotter. (and this is one reason why I suggested a TO-220 device. Even if your switching losses get quite high, it's not going to get too hot)
There is another problem. As Bob suggests, don't go trying to run this at 100kHz -- why?
lets do some math... At 100kHz, you are trying to turn the device off 100,000 times per second. But that device takes 1/100,000 of a second to turn off! This means that the device will never get to turn off before its asked to turn back on again. If your circuit relies on the device turning off, it's not going to work.
So that's the second problem. You may not even be able to turn it off completely.
This is also the reason why I told you not to go chasing very low Rds(on) values. As you can see, the Rds(on), once fairly low, contributes minimally to your power dissipation -- especially at low currents (1A in this case is a very low current!).
Rds(on) is reduced, essentially, at the cost of increasing gate capacitance. And it is that gate capacitance which contributes to slower switching speeds. And as you've seen, the losses during switching can be significant.
I guess you're beginning to see that the answer to "what mosfet should I use" is rather a complex one, and one where the person assisting may need to make assumptions about your project or ask questions that seem strange.
I think I assumed the switching speed was low... How did I do that and was it justified?
Ah.. OK, your question included the following:
I am looking for a P Channel, Mosfet which will be turned ON for 10 - 100mS and off for 30 minutes.
If you had said 2uS and off for 10uS, the solution would have been significantly different and we would have been talking about gate drivers for your mosfet.