A great site for info on these kinds of things is s4gru.com. But the answer is yes. Anytime your phone has to switch radios, look for signal, etc. It’s going to negatively impact your battery. If you’ve ever been somewhere where your phone was constantly searching for signal (had this happen with my old Samsung A900 constantly) it can chew through your battery in an hour or less sometimes.
Since Sprint’s LTE is still spotty in a lot of areas the constant switch from LTE to EVDO can definitely lessen your battery life. On the HTC One the EVDO Radio is on the bottom of the device and the LTE radio is on the top so it actually has to switch radios depending on which signal it is receiving. Not to mention Sprint’s LTE Band 25 is on the 1900Mhz spectrum which has worse object penetration than AT&T’s and Verizon’s LTE.
On the other hand my location was one of the first areas to get LTE with Sprint and I live in the downtown area and I believe LTE rollout is at 70%-80% complete. I get constant LTE connection and my battery life has been fantastic on the HTC One typically getting 14 hours of battery time by the time it hits 20%. This is with heavy use since I just got the device and have been setting it up and getting use to Android again after coming from WP7 after almost 3 years.
Under common law HTC should be able to recover from ST on several different contract claims. Amsterdam is a civil law jurisdiction though, but I’m sure they still have laws in place that would allow HTC to recover for loss of sales of the device, etc.
On a side note just switched from an HTC Arrive to the One and the audio output and recording abilities of the device are impressive. I hope Nokia, ST, and HTC can work this out quickly so that Nokia can get just compensation for their hard work and HTC can continue to sell an excellent device.